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

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

Proof:
{{begin-eqn}}
{{eqn | l = \csc 105 \degrees
      | r = \map \csc {180 \degrees - 75 \degrees}
      | c = 
}}
{{eqn | r = \csc 75 \degrees
      | c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 6 - \sqrt 2
      | c = Cosecant of $75 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 120 Degrees

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

Proof:
{{begin-eqn}}
{{eqn | l = \csc 120 \degrees
      | r = \map \csc {180 \degrees - 60 \degrees}
      | c = 
}}
{{eqn | r = \csc 60 \degrees
      | c = Cosecant of Supplementary Angle
}}
{{eqn | r = \frac {2 \sqrt 3} 3
      | c = Cosecant of $60 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 135 Degrees

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

Proof:
{{begin-eqn}}
{{eqn | l = \csc 135 \degrees
      | r = \map \csc {180 \degrees - 45 \degrees}
      | c = 
}}
{{eqn | r = \csc 45 \degrees
      | c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 2
      | c = Cosecant of $45 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 150 Degrees

:$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 150 \degrees
      | r = \map \csc {180 \degrees - 30 \degrees}
      | c = 
}}
{{eqn | r = \csc 30 \degrees
      | c = Cosecant of Supplementary Angle
}}
{{eqn | r = 2
      | c = Cosecant of $30 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 165 Degrees

:$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 165 \degrees
      | r = \map \csc {180 \degrees - 15 \degrees}
      | c = 
}}
{{eqn | r = \csc 15 \degrees
      | c = Cosecant of Supplementary Angle
}}
{{eqn | r = \sqrt 6 + \sqrt 2
      | c = Cosecant of 15 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Straight Angle

:$\csc 180 \degrees = \csc \pi$ is undefined

Proof:
From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Straight Angle:
:$\sin \pi = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}}
Theorem: Cosecant of 195 Degrees

:$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 195^\circ
      | r = \csc \left({360^\circ - 165^\circ}\right)
      | c = 
}}
{{eqn | r = -\csc 165^\circ
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = - \left({\sqrt 6 + \sqrt 2}\right)
      | c = Cosecant of 165 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 210 Degrees

:$\csc 210 \degrees = \csc \dfrac {7 \pi} 6 = -2$

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \csc 240 \degrees
      | r = \map \csc {360 \degrees - 120 \degrees}
      | c = 
}}
{{eqn | r = -\csc 120 \degrees
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
      | c = Cosecant of $120 \degrees$
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 255 Degrees

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

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

:$\csc 270 \degrees = \csc \dfrac {3 \pi} 2 = -1$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 270 \degrees
      | r = \map \csc {360 \degrees - 90 \degrees}
      | c = 
}}
{{eqn | r = -\csc 90 \degrees
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = -1
      | c = Cosecant of Right Angle
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 285 Degrees

:$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

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

:$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 300^\circ
      | r = \csc \left({360^\circ - 60^\circ}\right)
      | c = 
}}
{{eqn | r = -\csc 60^\circ
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\frac {2 \sqrt 3} 3
      | c = Cosecant of 60 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 315 Degrees

:$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 315^\circ
      | r = \csc \left({360^\circ - 45^\circ}\right)
      | c = 
}}
{{eqn | r = -\csc 45^\circ
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = -\sqrt 2
      | c = Cosecant of 45 Degrees
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 330 Degrees

:$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$

Proof:
{{begin-eqn}}
{{eqn | l = \csc 330 \degrees
      | r = \map \csc {360 \degrees - 30 \degrees}
      | c = 
}}
{{eqn | r = -\csc 30 \degrees
      | c = Cosecant of Conjugate Angle
}}
{{eqn | r = -2
      | c = {{csc|30}}
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of 345 Degrees

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

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

:$\csc 360 \degrees = \csc 2 \pi$ is undefined

Proof:
From Cosecant is Reciprocal of Sine:
:$\csc \theta = \dfrac 1 {\sin \theta}$
From Sine of Full Angle:
:$\sin 360 \degrees = 0$
Thus $\csc \theta$ is undefined at this value.
{{qed}}
Theorem: Cosecant Function is Odd

:$\map \csc {-x} = -\csc x$
That is, the cosecant function is odd.

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {-x}
      | r = \frac 1 {\map \sin {-x} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin x}
      | c = Sine Function is Odd
}}
{{eqn | r = -\csc x
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant Function is Even

:$\map \sec {-x} = \sec x$
That is, the secant function is even.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sech {-x}
      | r = \frac 1 {\map \cosh {-x} } 
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac 1 {\cosh x} 
      | c = Hyperbolic Cosine Function is Even
}}
{{eqn | r = \sech x
}}
{{end-eqn}}
{{qed}}
Theorem: Secant Function is Even

:$\map \sec {-x} = \sec x$
That is, the secant function is even.

Proof:
{{begin-eqn}}
{{eqn	| l = \sech \paren {-x}
	| r = \frac 1 {\cosh \paren {-x} }
	| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn	| r = \frac 1 {\cos \paren {-i x} }
	| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn	| r = \frac 1 {\cos \paren {i x} }
	| c = Cosine Function is Even
}}
{{eqn	| r = \frac 1 {\cosh x}
	| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn	| r = \sech x
	| c = {{Defof|Hyperbolic Secant}}
}}
{{end-eqn}}
{{qed}}
Theorem: Secant Function is Even

:$\map \sec {-x} = \sec x$
That is, the secant function is even.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {-x}
      | r = \frac 1 {\map \cos {-x} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\cos x}
      | c = Cosine Function is Even
}}
{{eqn | r = \sec x
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent Function is Odd

:$\map \cot {-x} = -\cot x$
That is, the cotangent function is odd.

Proof:
{{begin-eqn}}
{{eqn | l = \map \cot {-x}
      | r = \frac {\map \cos {-x} } {\map \sin {-x} }
      | c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\sin x} {\cos x}
      | c = Cosine Function is Even and Sine Function is Odd
}}
{{eqn | r = -\cot x
      | c = Cotangent is Cosine divided by Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Sum

:$\map \cot {a + b} = \dfrac {\cot a \cot b - 1} {\cot b + \cot a}$
where $\cot $ is cotangent.
=== Corollary ===
{{:Cotangent of Sum/Corollary}}

Proof:
{{begin-eqn}}
{{eqn | l = \map \cot {a + b}
      | r = \frac {\map \cos {a + b} } {\map \sin {a + b} } 
      | c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\cos a \cos b - \sin a \sin b} {\sin a \cos b + \cos a \sin b}
      | c = Cosine of Sum and Sine of Sum
}}
{{eqn | r = \frac {\frac {\cos a \cos b} {\sin a \sin b} - 1} {\frac {\cos b} {\sin b} + \frac {\cos a} {\sin a} }
      | c = dividing numerator and denominator by $\sin a \sin b$
}}
{{eqn | r = \frac {\cot a \cot b - 1} {\cot b + \cot a}
      | c = Cotangent is Cosine divided by Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Half-Integer is Half Odd Integer

Let $r$ be a number.
Then $r$ is a half-integer {{iff}} $r = \dfrac n 2$ where $n$ is an odd integer.

Proof:
=== Necessary Condition ===
Let $r$ be a half-integer.
Then by definition $r = n + \dfrac 1 2$ for some $n \in \Z$.
Thus:
{{begin-eqn}}
{{eqn | l = 2 r
      | r = 2 \paren {n + \dfrac 1 2}
      | c = 
}}
{{eqn | r = 2 n + 2 \paren {\frac 1 2}
      | c = 
}}
{{eqn | r = 2 n + 1
      | c = 
}}
{{end-eqn}}
thus showing that $r$ is half of $2 n + 1$ for some $n \in \Z$.
By Odd Integer 2n + 1 it follows that $r$ is half of an odd integer.
{{qed|lemma}}
Theorem: Sine of Angle plus Straight Angle

:$\map \sin {x + \pi} = -\sin x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sin {x + \pi}
      | r = \sin x \cos \pi + \cos x \sin \pi
      | c = Sine of Sum
}}
{{eqn | r = \sin x \cdot \paren {-1} + \cos x \cdot 0
      | c = Cosine of Straight Angle and Sine of Straight Angle
}}
{{eqn | r = -\sin x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Straight Angle

:$\map \cos {x + \pi} = -\cos x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
      | r = \cos x \cos \pi - \sin x \sin \pi
      | c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot \paren {-1} - \sin x \cdot 0
      | c = Cosine of Straight Angle and Sine of Straight Angle
}}
{{eqn | r = -\cos x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Straight Angle

:$\map \cos {x + \pi} = -\cos x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
      | r = \map \Re {\map \cos {x + \pi} + i \, \map \sin {x + \pi} }
      | c = 
}}
{{eqn | r = \map \Re {e^{i \paren {x + \pi} } }
      | c = Euler's Formula
}}
{{eqn | r = \map \Re {e^{i x + i \pi} }
      | c = 
}}
{{eqn | r = \map \Re {e^{i x} e^{i \pi} }
      | c = Exponential of Sum: Complex Numbers
}}
{{eqn | r = \map \Re {-e^{i x} }
      | c = Euler's Identity
}}
{{eqn | r = -\map \Re {e^{i x} }
      | c = 
}}
{{eqn | r = -\map \Re {\cos x + i \cos x}
      | c = Euler's Formula
}}
{{eqn | r = -\cos x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Straight Angle

:$\map \cos {x + \pi} = -\cos x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {x + \pi}
      | r = \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }
      | c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 2 \paren {e^{i x} e^{i \pi} + e^{-i x} e^{-i \pi} }
      | c = Exponential of Sum: Complex Numbers
}}
{{eqn | r = \frac 1 2 \paren {-e^{i x} - e^{-i x} }
      | c = Euler's Identity
}}
{{eqn | r = -\frac 1 2 \paren {e^{i x} + e^{-i x} }
      | c = 
}}
{{eqn | r = -\cos x
      | c = Euler's Cosine Identity
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Straight Angle

:$\map \cos {x + \pi} = -\cos x$

Proof:
From the discussion in the proof of Real Cosine Function is Periodic:
:$\map \sin {x + \eta} = \cos x$
:$\map \cos {x + \eta} = -\sin x$
for $\eta \in \R_{>0}$.
From Sine and Cosine are Periodic on Reals: Pi, we define $\pi \in \R$ as $\pi := 2 \eta$.
It follows that $\eta = \dfrac \pi 2$, thus:
:$\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$
{{qed}}
Theorem: Sine of Angle plus Full Angle

:$\map \sin {x + 2 \pi} = \sin x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sin {x + 2 \pi}
      | r = \sin x \cos 2 \pi + \cos x \sin 2 \pi
      | c = Sine of Sum
}}
{{eqn | r = \sin x \cdot 1 + \cos x \cdot 0
      | c = Cosine of Full Angle and Sine of Full Angle
}}
{{eqn | r = \sin x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Full Angle

:$\map \cos {x + 2 \pi} = \cos x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {x + 2 \pi}
      | r = \cos x \cos 2 \pi - \sin x \sin 2 \pi
      | c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot 1 - \sin x \cdot 0
      | c = Cosine of Full Angle and Sine of Full Angle
}}
{{eqn | r = \cos x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Tangent of Angle plus Full Angle

:$\map \tan {x + 2 \pi} = \tan x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \tan {x + 2 \pi}
      | r = \frac {\map \sin {x + 2 \pi} } {\map \cos {x + 2 \pi} } 
      | c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\sin x} {\cos x}
      | c = Sine of Angle plus Full Angle and Cosine of Angle plus Full Angle
}}
{{eqn | r = \tan x
      | c = Tangent is Sine divided by Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Tangent of Angle plus Straight Angle

: $\tan \left({x + \pi}\right) = \tan x$

Proof:
{{begin-eqn}}
{{eqn | l = \tan \left({x + \pi}\right)
      | r = \frac {\sin \left({x + \pi}\right)} {\cos \left({x + \pi}\right)} 
      | c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\sin x} {-\cos x}
      | c = Sine of Angle plus Straight Angle and Cosine of Angle plus Straight Angle
}}
{{eqn | r = \tan x
      | c = Tangent is Sine divided by Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Angle plus Straight Angle

:$\map \cot {x + \pi} = \cot x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cot {x + \pi}
      | r = \frac {\map \cos {x + \pi} } {\map \sin {x + \pi} }
      | c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {-\cos x} {-\sin x}
      | c = Cosine of Angle plus Straight Angle and Sine of Angle plus Straight Angle
}}
{{eqn | r = \cot x
      | c = Cotangent is Cosine divided by Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Angle plus Straight Angle

:$\map \sec {x + \pi} = -\sec x$

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

:$\map \csc {x + \pi} = -\csc x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {x + \pi}
      | r = \frac 1 {\map \sin {x + \pi} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\sin x}
      | c = Sine of Angle plus Straight Angle
}}
{{eqn | r = -\csc x
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Angle plus Full Angle

: $\cot \left({x + 2 \pi}\right) = \cot x$

Proof:
{{begin-eqn}}
{{eqn | l = \cot \left({x + 2 \pi}\right)
      | r = \frac {\cos \left({x + 2 \pi}\right)} {\sin \left({x + 2 \pi}\right)}
      | c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\cos x} {\sin x}
      | c =  Cosine of Angle plus Full Angle and Sine of Angle plus Full Angle
}}
{{eqn | r = \cot x
      | c = Cotangent is Cosine divided by Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Angle plus Full Angle

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

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

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

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

:$\map \sin {x + \dfrac {3 \pi} 2} = -\cos x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sin {x + \frac {3 \pi} 2}
      | r = \sin x \cos \frac {3 \pi} 2 + \cos x \sin \frac {3 \pi} 2
      | c = Sine of Sum
}}
{{eqn | r = \sin x \cdot 0 + \cos x \cdot \paren {-1}
      | c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
{{eqn | r = -\cos x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Angle plus Three Right Angles

:$\map \cos {x + \dfrac {3 \pi} 2} = \sin x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {x + \frac {3 \pi} 2}
      | r = \cos x \cos \frac {3 \pi} 2 - \sin x \sin \frac {3 \pi} 2
      | c = Cosine of Sum
}}
{{eqn | r = \cos x \cdot 0 - \sin x \cdot \paren {-1}
      | c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
{{eqn | r = \sin x
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Tangent of Angle plus Three Right Angles

:$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \tan {x + \frac {3 \pi} 2}
      | r = \frac {\map \sin {x + \frac {3 \pi} 2} } {\map \cos {x + \frac {3 \pi} 2} } 
      | c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\cos x} {\sin x}
      | c = Sine of Angle plus Three Right Angles and Cosine of Angle plus Three Right Angles
}}
{{eqn | r = -\cot x
      | c = Cotangent is Cosine divided by Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cotangent of Angle plus Three Right Angles

:$\map \cot {x + \dfrac {3 \pi} 2} = -\tan x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \cot {x + \frac {3 \pi} 2}
      | r = \frac {\map \cos {x + \frac {3 \pi} 2} } {\map \sin {x + \frac {3 \pi} 2} }
      | c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\sin x} {-\cos x}
      | c = Cosine of Angle plus Right Angle and Sine of Angle plus Right Angle
}}
{{eqn | r = -\tan x
      | c = Tangent is Sine divided by Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Angle plus Three Right Angles

:$\map \sec {x + \dfrac {3 \pi} 2} = \csc x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {x + \frac {3 \pi} 2}
      | r = \frac 1 {\map \cos {x + \frac {3 \pi} 2} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {\sin x}
      | c = Cosine of Angle plus Three Right Angles
}}
{{eqn | r = \csc x
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Angle plus Three Right Angles

:$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {x + \frac {3 \pi} 2}
      | r = \frac 1 {\map \sin {x + \frac {3 \pi} 2} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\cos x}
      | c = Sine of Angle plus Three Right Angles
}}
{{eqn | r = -\sec x
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Sine of Three Right Angles less Angle

:$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$
where $\sin$ and $\cos$ are sine and cosine respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sin {\frac {3 \pi} 2 - \theta}
      | r = \sin \frac {3 \pi} 2 \cos \theta - \cos \frac {3 \pi} 2 \sin \theta
      | c = Sine of Difference
}}
{{eqn | r = \paren {-1} \times \cos \theta - 0 \times \sin \theta
      | c = Sine of Three Right Angles and Cosine of Three Right Angles
}}
{{eqn | r = -\cos \theta
}}
{{end-eqn}}
{{qed}}
Theorem: Cosine of Three Right Angles less Angle

:$\map \cos {\dfrac {3 \pi} 2 - \theta} = -\sin \theta$
where $\cos$ and $\sin$ are cosine and sine respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \cos {\frac {3 \pi} 2 - \theta}
      | r = \cos \frac {3 \pi} 2 \cos \theta + \sin \frac {3 \pi} 2 \sin \theta
      | c = Cosine of Difference
}}
{{eqn | r = 0 \times \cos \theta + \paren {-1} \times \sin \theta
      | c = Cosine of Three Right Angles and Sine of Three Right Angles
}}
{{eqn | r = -\sin \theta
}}
{{end-eqn}}
{{qed}}
Theorem: Secant of Three Right Angles less Angle

:$\map \sec {\dfrac {3 \pi} 2 - \theta} = -\csc \theta$
where $\sec$ and $\csc$ are secant and cosecant respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \sec {\frac {3 \pi} 2 - \theta}
      | r = \frac 1 {\map \cos {\frac {3 \pi} 2 - \theta} }
      | c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \frac 1 {-\sin \theta}
      | c = Cosine of Three Right Angles less Angle
}}
{{eqn | r = -\csc \theta
      | c = Cosecant is Reciprocal of Sine
}}
{{end-eqn}}
{{qed}}
Theorem: Cosecant of Three Right Angles less Angle

:$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$
where $\csc$ and $\sec$ are cosecant and secant respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \map \csc {\frac {3 \pi} 2 - \theta}
      | r = \frac 1 {\map \sin {\frac {3 \pi} 2 - \theta} }
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \frac 1 {-\cos \theta}
      | c = Sine of Three Right Angles less Angle
}}
{{eqn | r = -\sec \theta
      | c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
{{qed}}
Theorem: Triple Angle Formulas/Sine

:$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$

Proof:
{{begin-eqn}}
{{eqn | l = \sin 3 \theta
      | r = \map \sin {2 \theta + \theta}
}}
{{eqn | r = \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta
      | c = Sine of Sum
}}
{{eqn | r = \paren {2 \sin \theta \cos \theta} \cos \theta + \paren {\cos^2 \theta - \sin^2 \theta} \sin \theta
      | c = Double Angle Formula for Sine and Double Angle Formula for Cosine
}}
{{eqn | r = 2 \sin \theta \cos^2 \theta + \cos^2 \theta \sin \theta - \sin^3 \theta
      | c = 
}}
{{eqn | r = 2 \sin \theta \paren {1 - \sin^2 \theta} + \paren {1 - \sin^2 \theta} \sin \theta - \sin^3 \theta
      | c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - \sin^3 \theta - \sin^3 \theta
      | c = multiplying out
}}
{{eqn | r = 3 \sin \theta - 4 \sin^3 \theta
      | c = gathering terms
}}
{{end-eqn}}
{{qed}}
Theorem: Triple Angle Formulas/Sine

:$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$

Proof:
We have:
{{begin-eqn}}
{{eqn | l = \cos 3 \theta + i \sin 3 \theta
      | r = \paren {\cos \theta + i \sin \theta}^3
      | c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}
}}
{{eqn | o = 
      | ro=+
      | r = \binom 3 2 \paren {\cos \theta} \paren {i \sin \theta}^2 + \paren {i \sin \theta}^3
      | c = Binomial Theorem
}}
{{eqn | r = \cos^3 \theta + 3 i \cos^2 \theta \sin \theta + 3 i^2 \cos \theta \sin^2 \theta + i^3 \sin^3 \theta
      | c = substituting for binomial coefficients
}}
{{eqn | r = \cos^3 \theta + 3 i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta
      | c = $i^2 = -1$
}}
{{eqn | n = 1
      | r = \cos^3 \theta - 3 \cos \theta \sin^2 \theta 
}}
{{eqn | o = 
      | ro=+
      | r = i \paren {3 \cos^2 \theta \sin \theta - \sin^3 \theta}
      | c = rearranging
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \sin 3 \theta
      | r = 3 \cos^2 \theta \sin \theta - \sin^3 \theta
      | c = equating imaginary parts in $(1)$
}}
{{eqn | r = 3 \paren {1 - \sin^2 \theta} \sin \theta - \sin^3 \theta
      | c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = 3 \sin \theta - 4 \sin^3 \theta
      | c = multiplying out and gathering terms
}}
{{end-eqn}}
{{qed}}