avewright/proofwiki-math huggingface dataset:
Theorem: Cotangent of 105 Degrees

:$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 105^\circ
      | r = \cot \left({90^\circ + 15^\circ}\right)
      | c = 
}}
{{eqn | r = - \tan 15^\circ
      | c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = - \left({2 - \sqrt 3}\right)
      | c = Tangent of 15 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 120 Degrees

:$\cot 120 \degrees = \cot \dfrac {2 \pi} 3 = -\dfrac {\sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 120 \degrees
      | r = \map \cot {90 \degrees + 30 \degrees}
      | c = 
}}
{{eqn | r = -\tan 30 \degrees
      | c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\frac {\sqrt 3} 3
      | c = Tangent of 30 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 135 Degrees

:$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 135 \degrees
      | r = \cot \paren {90 \degrees + 45 \degrees}
      | c = 
}}
{{eqn | r = -\tan 45 \degrees
      | c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = - 1
      | c = Tangent of $45 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 150 Degrees

:$\cot 150 \degrees = \cot \dfrac {5 \pi} 6 = -\sqrt 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 150 \degrees
      | r = \map \cot {90 \degrees + 60 \degrees}
      | c = 
}}
{{eqn | r = -\tan 60 \degrees
      | c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\sqrt 3
      | c = Tangent of $60 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 165 Degrees

:$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 165 \degrees
      | r = \map \cot {90 \degrees + 75 \degrees}
      | c = 
}}
{{eqn | r = -\tan 75 \degrees
      | c = Cotangent of Angle plus Right Angle
}}
{{eqn | r = -\paren {2 + \sqrt 3}
      | c = {{tan|75}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Straight Angle

:$\cot 180^\circ = \cot \pi$ is undefined

Proof:
From Cotangent is Cosine divided by Sine:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
When $\sin \theta = 0$, $\dfrac {\cos \theta} {\sin \theta}$ can be defined only if $\cos \theta = 0$.
But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$.
When $\theta = \pi$, $\sin \theta = 0$.
Thus $\cot \theta$ is undefined at this value.
{{qed}}
Theorem: Cotangent of 195 Degrees

:$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 195 \degrees
      | r = \map \cot {360 \degrees - 165 \degrees}
      | c = 
}}
{{eqn | r = -\cot 165 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = 2 + \sqrt 3
      | c = {{cot|165}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 210 Degrees

:$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 210^\circ
      | r = \cot \left({360^\circ - 150^\circ}\right)
      | c = 
}}
{{eqn | r = -\cot 150^\circ
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = \sqrt 3
      | c = Cotangent of 150 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 225 Degrees

:$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 225^\circ
      | r = \cot \left({360^\circ - 135^\circ}\right)
      | c = 
}}
{{eqn | r = -\cot 135^\circ
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = 1
      | c = Cotangent of 135 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 240 Degrees

:$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 240^\circ
      | r = \cot \left({360^\circ - 120^\circ}\right)
      | c = 
}}
{{eqn | r = -\cot 120^\circ
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = \frac {\sqrt 3} 3
      | c = Cotangent of 120 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 255 Degrees

:$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 255 \degrees
      | r = \map \cot {360 \degrees - 105 \degrees}
      | c = 
}}
{{eqn | r = -\cot 105 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = 2 - \sqrt 3
      | c = Cotangent of $105 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Three Right Angles

:$\cot 270 \degrees = \cot \dfrac {3 \pi} 2 = 0$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 270 \degrees
      | r = \map \cot {360 \degrees - 90 \degrees}
      | c = 
}}
{{eqn | r = -\cot 90 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = 0
      | c = Cotangent of Right Angle
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 285 Degrees

:$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 285 \degrees
      | r = \cot \paren {360 \degrees - 75 \degrees}
      | c = 
}}
{{eqn | r = -\cot 75^\circ
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\paren {2 - \sqrt 3}
      | c = Cotangent of $75 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 300 Degrees

:$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 300 \degrees
      | r = \map \cot {360 \degrees - 60 \degrees}
      | c = 
}}
{{eqn | r = -\cot 60 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\frac {\sqrt 3} 3
      | c = {{cot|60}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 315 Degrees

:$\cot 315 \degrees = \cot \dfrac {7 \pi} 4 = -1$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 315 \degrees
      | r = \map \cot {360 \degrees - 45 \degrees}
      | c = 
}}
{{eqn | r = -\cot 45 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = -1
      | c = Cotangent of $45 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 330 Degrees

:$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 330^\circ
      | r = \cot \left({360^\circ - 30^\circ}\right)
      | c = 
}}
{{eqn | r = -\cot 30^\circ
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\sqrt 3
      | c = Cotangent of 30 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of 345 Degrees

:$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$

Proof:
{{begin-eqn}}
{{eqn | l = \cot 345 \degrees
      | r = \map \cot {360 \degrees - 15 \degrees}
      | c = 
}}
{{eqn | r = -\cot 15 \degrees
      | c = Cotangent of Conjugate Angle
}}
{{eqn | r = -\paren {2 + \sqrt 3}
      | c = Cotangent of $15 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Full Angle

:$\cot 360^\circ = \cot 2 \pi$ is undefined

Proof:
From Cotangent is Cosine divided by Sine:
: $\cot \theta = \dfrac {\cos \theta} {\sin \theta}$
From Cosine of Full Angle:
: $\cos 2 \pi = 1$
From Sine of Full Angle:
: $\sin 2 \pi = 0$
Thus $\cot \theta$ is undefined at this value.
{{qed}}
Theorem: Secant of Zero

:$\sec 0 = 0$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 0 \degrees
      | r = \frac 1 {\cos 0 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 1
      | c = Cosine of Zero is One
}}
{{eqn | r = 1
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 15 Degrees

:$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 15 \degrees
      | r = \frac 1 {\cos 15 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
      | c = {{cos|15}}
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }
      | c = multiplying top and bottom by $\sqrt 6 - \sqrt 2$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {6 - 2}
      | c = Difference of Two Squares
}}
{{eqn | r = \sqrt 6 - \sqrt 2
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 30 Degrees

:$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 30 \degrees
      | r = \frac 1 {\cos 30 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
      | c = {{cos|30}}
}}
{{eqn | r = \frac {2 \sqrt 3} 3
      | c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 45 Degrees

:$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 45 \degrees
      | r = \frac 1 {\cos 45 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
      | c = Cosine of $45 \degrees$
}}
{{eqn | r = \sqrt 2
      | c = multiplying top and bottom by $2 \sqrt 2$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 60 Degrees

:$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 60 \degrees
      | r = \frac 1 {\cos 60 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac 1 2}
      | c = Cosine of $60 \degrees$
}}
{{eqn | r = 2
      | c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 75 Degrees

:$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 75 \degrees
      | r = \frac 1 {\cos 75 \degrees}
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
      | c = {{cos|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
      | c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 + \sqrt 2} } {\paren {\sqrt 6 - \sqrt 2} \paren {\sqrt 6 + \sqrt 2} }
      | c = multiplying top and bottom by $\sqrt 6 + \sqrt 2$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 + \sqrt 2} } {6 - 2}
      | c = Difference of Two Squares
}}
{{eqn | r = \sqrt 6 + \sqrt 2
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Right Angle

:$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined

Proof:
From Secant is Reciprocal of Cosine:
:$\sec \theta = \dfrac 1 {\cos \theta}$
From Cosine of Right Angle:
:$\cos \dfrac \pi 2 = 0$
Thus $\sec \theta$ is undefined at this value.
{{qed}}
Theorem: Secant of Angle plus Right Angle

: $\sec \left({x + \dfrac \pi 2}\right) = -\csc x$

Proof:
{{begin-eqn}}
{{eqn | l = \sec \left({x + \frac \pi 2}\right)
      | r = \frac 1 {\cos \left({x + \frac \pi 2}\right)} 
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {- \sin x}
      | c = Cosine of Angle plus Right Angle
}}
{{eqn | r = -\csc x
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Angle plus Right Angle

: $\csc \left({x + \dfrac \pi 2}\right) = \sec x$

Proof:
{{begin-eqn}}
{{eqn | l = \csc \left({x + \frac \pi 2}\right)
      | r = \frac 1 {\sin \left({x + \frac \pi 2}\right)} 
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\cos x}
      | c = Sine of Angle plus Right Angle
}}
{{eqn | r = \sec x
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Complement equals Cosecant

:$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$
where $\sec$ and $\csc$ are secant and cosecant respectively.
That is, the cosecant of an angle is the secant of its complement.
This relation is defined wherever $\sin \theta \ne 0$.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {\frac \pi 2 - \theta}
      | r = \frac 1 {\map \cos {\frac \pi 2 - \theta} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\sin \theta}
      | c = Cosine of Complement equals Sine
}}
{{eqn | r = \csc \theta
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
The above is valid only where $\sin \theta \ne 0$, as otherwise $\dfrac 1 {\sin \theta}$ is undefined.
From Sine of Multiple of Pi it follows that this happens when $\theta \ne n \pi$.
{{qed}}
Theorem: Cosecant of Complement equals Secant

:$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$
where $\csc$ and $\sec$ are cosecant and secant respectively.
That is, the secant of an angle is the cosecant of its complement.
This relation is defined wherever $\cos \theta \ne 0$.

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {\frac \pi 2 - \theta}
      | r = \frac 1 {\map \sin {\frac \pi 2 - \theta} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\cos \theta}
      | c = Sine of Complement equals Cosine
}}
{{eqn | r = \sec \theta
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
The above is valid only where $\cos \theta \ne 0$, as otherwise $\dfrac 1 {\cos \theta}$ is undefined.
From Cosine of Half-Integer Multiple of Pi it follows that this happens when $\theta \ne \paren {2 n + 1} \dfrac \pi 2$.
{{qed}}
Theorem: Secant of Supplementary Angle

:$\map \sec {\pi - \theta} = -\sec \theta$
where $\sec$ denotes secant.
That is, the secant of an angle is the negative of its supplement.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {\pi - \theta}
      | r = \frac 1 {\map \cos {\pi - \theta} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {-\cos \theta} 
      | c = Cosine of Supplementary Angle
}}
{{eqn | r = -\sec \theta
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Supplementary Angle

:$\map \csc {\pi - \theta} = \csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle equals its supplement.

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {\pi - \theta}
      | r = \frac 1 {\map \sin {\pi - \theta} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\sin \theta} 
      | c = Sine of Supplementary Angle
}}
{{eqn | r = \csc \theta
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Conjugate Angle

:$\map \sec {2 \pi - \theta} = \sec \theta$
where $\sec$ denotes secant.
That is, the secant of an angle equals its conjugate.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {2 \pi - \theta}
      | r = \frac 1 {\map \cos {2 \pi - \theta} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\cos \theta}
      | c = Cosine of Conjugate Angle
}}
{{eqn | r = \sec \theta
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Conjugate Angle

:$\map \csc {2 \pi - \theta} = -\csc \theta$
where $\csc$ denotes cosecant.
That is, the cosecant of an angle is the negative of its conjugate.

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {2 \pi - \theta}
      | r = \frac 1 {\map \sin {2 \pi - \theta} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin \theta}
      | c = Cosine of Conjugate Angle and Sine of Conjugate Angle
}}
{{eqn | r = -\csc \theta
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 105 Degrees

:$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 105 \degrees
      | r = \map \sec {90 \degrees + 15 \degrees}
      | c = 
}}
{{eqn | r = -\csc 15 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
      | c = Cosecant of $15 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 120 Degrees

:$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 120 \degrees
      | r = \map \sec {90 \degrees + 30 \degrees}
      | c = 
}}
{{eqn | r = -\csc 30 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -2
      | c = Cosecant of $30 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 135 Degrees

:$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 135 \degrees
      | r = \map \sec {90 \degrees + 45 \degrees}
      | c = 
}}
{{eqn | r = -\csc 45 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\sqrt 2
      | c = Cosecant of $45 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 150 Degrees

:$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 150 \degrees
      | r = \map \sec {90 \degrees + 60 \degrees}
      | c = 
}}
{{eqn | r = -\csc 60 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\dfrac {2 \sqrt 3} 3
      | c = Cosecant of $60 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 165 Degrees

:$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 165 \degrees
      | r = \map \sec {90 \degrees + 75 \degrees}
      | c = 
}}
{{eqn | r = -\csc 75 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
      | c = Cosecant of $75 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Straight Angle

:$\sec 180 \degrees = \sec \pi = -1$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 180 \degrees
      | r = \map \sec {90 \degrees + 90 \degrees}
      | c = 
}}
{{eqn | r = -\csc 90 \degrees
      | c = Secant of Angle plus Right Angle
}}
{{eqn | r = -1
      | c = Cosecant of Right Angle
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 195 Degrees

:$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 195 \degrees
      | r = \map \sec {360 \degrees - 165 \degrees}
      | c = 
}}
{{eqn | r = \sec 165 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 - \sqrt 2}
      | c = Secant of $165 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 210 Degrees

:$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 210 \degrees
      | r = \map \sec {360 \degrees - 150 \degrees}
      | c = 
}}
{{eqn | r = \sec 150 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = -2 \frac {\sqrt 3} 3
      | c = Secant of $150 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 225 Degrees

:$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 225 \degrees
      | r = \map \sec {360 \degrees - 135 \degrees}
      | c = 
}}
{{eqn | r = \sec 135 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = -\sqrt 2
      | c = Secant of $135 degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 240 Degrees

:$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 240 \degrees
      | r = \map \sec {360 \degrees - 120 \degrees}
      | c = 
}}
{{eqn | r = \sec 120 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = -2
      | c = Secant of $120 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 255 Degrees

:$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 255 \degrees
      | r = \map \sec {360 \degrees - 105 \degrees}
      | c = 
}}
{{eqn | r = \sec 105 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = -\paren {\sqrt 6 + \sqrt 2}
      | c = Secant of $105 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Three Right Angles

:$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined

Proof:
{{begin-eqn}}
{{eqn | l = \sec 270 \degrees
      | r = \map \sec {360 \degrees - 90 \degrees}
      | c = 
}}
{{eqn | r = \sec 90 \degrees
      | c = Secant of Conjugate Angle
}}
{{end-eqn}}
But from Secant of Right Angle, $\sec 90 \degrees$ is undefined.
{{qed}}
Theorem: Secant of 285 Degrees

:$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 285 \degrees
      | r = \map \sec {360 \degrees - 75 \degrees}
      | c = 
}}
{{eqn | r = \sec 75 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 6 + \sqrt 2
      | c = Secant of $75 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 300 Degrees

:$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 300 \degrees
      | r = \map \sec {360 \degrees - 60 \degrees}
      | c = 
}}
{{eqn | r = \sec 60 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = 2
      | c = Secant of $60 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 315 Degrees

:$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 315 \degrees
      | r = \map \sec {360 \degrees - 45 \degrees}
      | c = 
}}
{{eqn | r = \sec 45 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 2
      | c = {{sec|45}}
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 330 Degrees

:$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 330 \degrees
      | r = \map \sec {360 \degrees - 30 \degrees}
      | c = 
}}
{{eqn | r = \sec 30 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = 2 \frac {\sqrt 3} 3
      | c = Secant of $30 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of 345 Degrees

:$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 345 \degrees
      | r = \map \sec {360 \degrees - 15 \degrees}
      | c = 
}}
{{eqn | r = \sec 15 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = \sqrt 6 - \sqrt 2
      | c = Secant of $15 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Full Angle

:$\sec 360 \degrees = \sec 2 \pi = 1$

Proof:
{{begin-eqn}}
{{eqn | l = \sec 360 \degrees
      | r = \map \sec {360 \degrees - 0 \degrees}
      | c = 
}}
{{eqn | r = \sec 0 \degrees
      | c = Secant of Conjugate Angle
}}
{{eqn | r = 1
      | c = Secant of Zero
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Zero

:$\csc 0$ is undefined

Proof:
From Cosecant is Reciprocal of Sine:
: $\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Zero is Zero:
: $\sin 0 = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}}
Theorem: Cosecant of 15 Degrees

:$\csc 15 \degrees = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 15 \degrees
      | r = \frac 1 {\sin 15 \degrees}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}
      | c = {{sin|15}}
}}
{{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}
      | c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 + \sqrt 2} } {\paren {\sqrt 6 - \sqrt 2} \paren {\sqrt 6 + \sqrt 2} }
      | c = multiplying top and bottom by $\sqrt 6 + \sqrt 2$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 + \sqrt 2} } {6 - 2}
      | c = Difference of Two Squares
}}
{{eqn | r = \sqrt 6 + \sqrt 2
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 30 Degrees

:$\csc 30^\circ = \csc \dfrac \pi 6 = 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 30^\circ
      | r = \frac 1 {\sin 30^\circ}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac 1 2}
      | c = Sine of $30^\circ$
}}
{{eqn | r = 2
      | c = multiplying top and bottom by $2$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 45 Degrees

:$\csc 45 \degrees = \csc \dfrac \pi 4 = \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 45 \degrees
      | r = \frac 1 {\sin 45 \degrees}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 2} 2}
      | c = {{sin|45}}
}}
{{eqn | r = \sqrt 2
      | c = multiplying top and bottom by $2 \sqrt 2$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 60 Degrees

:$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 60^\circ
      | r = \frac 1 {\sin 60^\circ}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 3} 2}
      | c = Sine of 60 Degrees
}}
{{eqn | r = \frac {2 \sqrt 3} 3
      | c = multiplying top and bottom by $2 \sqrt 3$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 75 Degrees

:$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 75^\circ
      | r = \frac 1 {\sin 75 \degrees}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {\frac {\sqrt 6 + \sqrt 2} 4}
      | c = {{sin|75}}
}}
{{eqn | r = \frac 4 {\sqrt 6 + \sqrt 2}
      | c = multiplying top and bottom by $4$
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }
      | c = multiplying top and bottom by $\sqrt 6 - \sqrt 2$ 
}}
{{eqn | r = \frac {4 \paren {\sqrt 6 - \sqrt 2} } {6 - 2}
      | c = Difference of Two Squares
}}
{{eqn | r = \sqrt 6 - \sqrt 2
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Right Angle

:$\csc 90 \degrees = \csc \dfrac \pi 2 = 1$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 90 \degrees
      | r = \frac 1 {\sin 90 \degrees}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 1
      | c = Sine of Right Angle
}}
{{eqn | r = 1
      | c = 
}}
{{end-eqn}}
{{qed}}