hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \cos 300 \degrees       | r = \map \cos {360 \degrees - 60 \degrees}       | c =  }} {{eqn | r = \cos 60 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = \frac 1 2       | c = Cosine of $60 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 315 \degrees       | r = \map \cos {360 \degrees - 45 \degrees}       | c =  }} {{eqn | r = \cos 45 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = \frac {\sqrt 2} 2       | c = Cosine of $45 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 330 \degrees       | r = \map \cos {360 \degrees - 30 \degrees}       | c =  }} {{eqn | r = \cos 30 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = \frac {\sqrt 3} 2       | c = Cosine of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos 345 \degrees       | r = \map \cos {360 \degrees - 15 \degrees}       | c =  }} {{eqn | r = \cos 15 \degrees       | c = Cosine of Conjugate Angle }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} 4       | c = Cosine of $15 \degrees$ }} {{end-eqn}} {{qed}}
A direct implementation of Cosine of Multiple of Pi: :$\forall n \in \Z: \cos n \pi = \paren {-1}^n$ In this case, $n = 2$ and so: :$\cos 2 \pi = \paren {-1}^2 = 1$ {{qed}}
{{begin-eqn}} {{eqn | l = \tan 0       | r = \frac {\sin 0} {\cos 0}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac 0 1       | c = Sine of Zero is Zero and Cosine of Zero is One }} {{eqn | r = 0       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 15^\circ       | r = \tan \frac {30^\circ} 2       | c =  }} {{eqn | r = \frac {\sin 30^\circ} {1 + \cos 30^\circ}       | c = Half Angle Formula for Tangent: Corollary 1 }} {{eqn | r = \frac {\frac 1 2} {1 + \frac {\sqrt 3} 2}       | c = Sine of $30^\circ$ and Cosine of $30^\circ$ }} {{eqn | r = \frac {\frac 1 2} {\frac {2 + \sqrt 3} 2}       | c =  }} {{eqn | r = \frac 1 {2 + \sqrt 3}       | c =  }} {{eqn | r = \frac {2 - \sqrt 3} {\left({2 + \sqrt 3}\right) \left({2 - \sqrt 3}\right)}       | c = multiplying top and bottom by $2 - \sqrt 3$ }} {{eqn | r = \frac {2 - \sqrt 3} {4 - 3}       | c = Difference of Two Squares }} {{eqn | r = 2 - \sqrt 3       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 15 \degrees       | r = \tan \frac {30 \degrees} 2       | c =  }} {{eqn | r = \frac {1 - \cos 30 \degrees} {\sin 30 \degrees}       | c = Half Angle Formula for Tangent: Corollary 2 }} {{eqn | r = \frac {1 - \frac {\sqrt 3} 2} {\frac 1 2}        | c = Cosine of $30 \degrees$ and Sine of $30 \degrees$ }} {{eqn | r = 2 - \sqrt 3       | c = multiplying top and bottom by $2$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 15^\circ       | r = \tan \frac {30^\circ} 2       | c =  }} {{eqn | r = \frac {\sin 15^\circ} {\cos 15^\circ}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}       | c = Sine of $15^\circ$ and Cosine of $15^\circ$ }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \sqrt 2}       | c = simplifying }} {{eqn | r = \frac {\left({\sqrt 6 - \sqrt 2}\right)^2} {\left({\sqrt 6 + \sqrt 2}\right) \left({\sqrt 6 - \sqrt 2}\right)}       | c = multiplying top and bottom by $\sqrt 6 - \sqrt 2$  }} {{eqn | r = \frac {6 - 2 \sqrt 6 \sqrt 2 + 2 } {6 - 2}       | c = multiplying out, and Difference of Two Squares }} {{eqn | r = \frac {8 - 4 \sqrt 3} 4       | c = simplifying }} {{eqn | r = 2 - \sqrt 3       | c = dividing top and bottom by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 30^\circ       | r = \frac {\sin 30^\circ} {\cos 30^\circ}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac 1 2} {\frac {\sqrt 3} 2}       | c = Sine of 30 Degrees and Cosine of 30 Degrees }} {{eqn | r = \frac 1 {\sqrt 3}       | c =  }} {{eqn | r = \frac {\sqrt 3} 3       | c = multiplying top and bottom by $\sqrt 3$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 45 \degrees       | r = \frac {\sin 45 \degrees} {\cos 45 \degrees}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}       | c = Sine of $45 \degrees$ and Cosine of $45 \degrees$ }} {{eqn | r = 1       | c = dividing top and bottom by $\sqrt 2 / 2$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 60^\circ       | r = \frac {\sin 60^\circ} {\cos 60^\circ}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac {\sqrt 3} 2} {\frac 1 2}       | c = Sine of 60 Degrees and Cosine of 60 Degrees }} {{eqn | r = \sqrt 3       | c = multiplying top and bottom by $2$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 75^\circ       | r = \frac {\sin 75^\circ} {\cos 75^\circ}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}       | c = Sine of 75 Degrees and Cosine of 75 Degrees }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - \sqrt 2}       | c = simplifying }} {{eqn | r = \frac {\left({\sqrt 6 + \sqrt 2}\right)^2} {\left({\sqrt 6 - \sqrt 2}\right) \left({\sqrt 6 + \sqrt 2}\right)}       | c = multiplying top and bottom by $\sqrt 6 + \sqrt 2$  }} {{eqn | r = \frac {6 + 2 \sqrt 6 \sqrt 2 + 2} {6 - 2}       | c = multiplying out, and Difference of Two Squares }} {{eqn | r = \frac {8 + 4 \sqrt 3} 4       | c = simplifying }} {{eqn | r = 2 + \sqrt 3       | c = dividing top and bottom by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \tan {\frac \pi 2 - \theta}       | r = \frac {\map \sin {\frac \pi 2 - \theta} } {\map \cos {\frac \pi 2 - \theta} }       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\cos \theta} {\sin \theta}       | c = Sine and Cosine of Complementary Angles }} {{eqn | r = \cot \theta       | c = Cotangent is Cosine divided by Sine }} {{end-eqn}} The above is valid only where $\sin \theta \ne 0$, as otherwise $\dfrac {\cos \theta} {\sin \theta}$ is undefined. From Sine of Multiple of Pi it follows that this happens when $\theta \ne n \pi$. {{qed}}
{{begin-eqn}} {{eqn | l=\cot \left({\frac \pi 2 - \theta}\right)       | r=\frac {\cos \left({\frac \pi 2 - \theta}\right)} {\sin \left({\frac \pi 2 - \theta}\right)}       | c=Cotangent is Cosine divided by Sine }} {{eqn | r=\frac {\sin \theta} {\cos \theta}       | c=Sine and Cosine of Complementary Angles }} {{eqn | r=\tan \theta       | c=Tangent is Sine divided by Cosine }} {{end-eqn}} The above is valid only where $\cos \theta \ne 0$, as otherwise $\dfrac {\sin \theta} {\cos \theta}$ is undefined. From Cosine of Half-Integer Multiple of Pi it follows that this happens when $\theta \ne \left({2 n + 1}\right) \dfrac \pi 2$. {{qed}}
{{begin-eqn}} {{eqn | l = \tan \left({x + \frac \pi 2}\right)       | r = \frac {\sin \left({x + \frac \pi 2}\right)} {\cos \left({x + \frac \pi 2}\right)}        | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\cos x} {- \sin x}       | c = Sine of Angle plus Right Angle and Cosine of Angle plus Right Angle }} {{eqn | r = -\cot x       | c = Cotangent is Cosine divided by Sine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot \left({x + \frac \pi 2}\right)       | r = \frac {\cos \left({x + \frac \pi 2}\right)}  {\sin \left({x + \frac \pi 2}\right)}       | 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}}
{{begin-eqn}} {{eqn | l = \map \tan {\pi - \theta}       | r = \frac {\map \sin {\pi - \theta} } {\map \cos {\pi - \theta} }       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {\sin \theta} {-\cos \theta}       | c = Sine of Supplementary Angle and Cosine of Supplementary Angle }} {{eqn | r = -\tan \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cot {\pi - \theta}       | r = \frac {\map \cos {\pi - \theta} } {\map \sin {\pi - \theta} }       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac {-\cos \theta} {\sin \theta}        | c = Cosine of Supplementary Angle and Sine of Supplementary Angle }} {{eqn | r = -\cot \theta }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \tan {2 \pi - \theta}       | r = \frac {\map \sin {2 \pi - \theta} } {\map \cos {2 \pi - \theta} }       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {-\sin \theta} {\cos \theta}       | c = Sine of Conjugate Angle and Cosine of Conjugate Angle }} {{eqn | r = -\tan \theta       | c = Tangent is Sine divided by Cosine }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map \cot {2 \pi - \theta}       | r = \frac {\map \cos {2 \pi - \theta} } {\map \sin {2 \pi - \theta} }       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac {\cos \theta} {-\sin \theta}       | c = Cosine of Conjugate Angle and Sine of Conjugate Angle }} {{eqn | r = -\cot \theta       | c = Cotangent is Cosine divided by Sine }} {{end-eqn}} {{qed}}
From Tangent is Sine divided by Cosine, $\tan \theta = \dfrac {\sin \theta} {\cos \theta}$. When $\cos \theta = 0$, $\dfrac {\sin \theta} {\cos \theta}$ can be defined only if $\sin \theta = 0$. But there are no such $\theta$ such that both $\cos \theta = 0$ and $\sin \theta = 0$. When $\theta = \dfrac \pi 2$, $\cos \theta = 0$. Thus $\tan \theta$ is undefined at this value. {{qed}}
{{begin-eqn}} {{eqn | l = \tan 105^\circ       | r = \tan \left({90^\circ + 15^\circ}\right)       | c =  }} {{eqn | r = - \cot 15^\circ       | c = Tangent of Angle plus Right Angle }} {{eqn | r = - \left({2 + \sqrt 3}\right)       | c = Cotangent of 15 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 120 \degrees       | r = \map \tan {90 \degrees + 30 \degrees}       | c =  }} {{eqn | r = -\cot 30 \degrees       | c = Tangent of Angle plus Right Angle }} {{eqn | r = -\sqrt 3       | c = Cotangent of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 135^\circ       | r = \tan \left({90^\circ + 45^\circ}\right)       | c =  }} {{eqn | r = - \cot 45^\circ       | c = Tangent of Angle plus Right Angle }} {{eqn | r = - 1       | c = Cotangent of 45 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 150^\circ       | r = \tan \left({90^\circ + 60^\circ}\right)       | c =  }} {{eqn | r = - \cot 60^\circ       | c = Tangent of Angle plus Right Angle }} {{eqn | r = - \dfrac {\sqrt 3} 3       | c = Cotangent of 60 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 165^\circ       | r = \tan \left({90^\circ + 75^\circ}\right)       | c =  }} {{eqn | r = - \cot 75^\circ       | c = Tangent of Angle plus Right Angle }} {{eqn | r = - \left({2 - \sqrt 3}\right)       | c = Cotangent of $75^\circ$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 180^\circ       | r = \frac {\sin 180^\circ} {\cos 180^\circ}       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac 0 {-1}       | c = Sine of Straight Angle and Cosine of Straight Angle }} {{eqn | r = 0       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 195^\circ       | r = \tan \left({360^\circ - 165^\circ}\right)       | c =  }} {{eqn | r = -\tan 165^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = 2 - \sqrt 3       | c = Tangent of 165 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 210 \degrees       | r = \map \tan {360 \degrees - 150 \degrees}       | c =  }} {{eqn | r = -\tan 150 \degrees       | c = Tangent of Conjugate Angle }} {{eqn | r = \frac {\sqrt 3} 3       | c = Tangent of $150 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 225 \degrees       | r = \map \tan {360 \degrees - 135 \degrees}       | c =  }} {{eqn | r = -\tan 135 \degrees       | c = Tangent of Conjugate Angle }} {{eqn | r = 1       | c = Tangent of $135 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 240 \degrees       | r = \map \tan {360 \degrees - 120 \degrees}       | c =  }} {{eqn | r = -\tan 120 \degrees       | c = Tangent of Conjugate Angle }} {{eqn | r = \sqrt 3       | c = Tangent of $120 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 255^\circ       | r = \tan \left({360^\circ - 105^\circ}\right)       | c =  }} {{eqn | r = -\tan 105^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = 2 + \sqrt 3       | c = Tangent of 105 Degrees }} {{end-eqn}} {{qed}}
We have: {{begin-eqn}} {{eqn | l = \tan 270^\circ       | r = \tan \left({360^\circ - 90^\circ}\right)       | c =  }} {{eqn | r = -\tan 90^\circ       | c = Tangent of Conjugate Angle }} {{end-eqn}} But from Tangent of Right Angle, $\tan 90^\circ$ is undefined. Hence so is $\tan 270^\circ$. {{qed}}
{{begin-eqn}} {{eqn | l = \tan 285^\circ       | r = \tan \left({360^\circ - 75^\circ}\right)       | c =  }} {{eqn | r = -\tan 75^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = -\left({2 + \sqrt 3}\right)       | c = Tangent of $75^\circ$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 300^\circ       | r = \tan \left({360^\circ - 60^\circ}\right)       | c =  }} {{eqn | r = -\tan 60^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = -\sqrt 3       | c = Tangent of 60 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 315^\circ       | r = \tan \left({360^\circ - 45^\circ}\right)       | c =  }} {{eqn | r = -\tan 45^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = -1       | c = Tangent of 60 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 330^\circ       | r = \tan \left({360^\circ - 30^\circ}\right)       | c =  }} {{eqn | r = -\tan 30^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = -\frac {\sqrt 3} 3       | c = Tangent of 30 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 345^\circ       | r = \tan \left({360^\circ - 15^\circ}\right)       | c =  }} {{eqn | r = -\tan 15^\circ       | c = Tangent of Conjugate Angle }} {{eqn | r = -\left({2 - \sqrt 3}\right)       | c = Tangent of 15 Degrees }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \tan 360^\circ       | r = \tan \left({360^\circ - 0^\circ}\right)       | c =  }} {{eqn | r = -\tan 0       | c = Tangent of Conjugate Angle }} {{eqn | r = 0       | c = Tangent of Zero }} {{end-eqn}} {{qed}}
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 $\theta = 0$, $\sin \theta = 0$. Thus $\cot \theta$ is undefined at this value. {{qed}}
{{begin-eqn}} {{eqn | l = \cot 30^\circ       | r = \frac {\cos 30^\circ} {\sin 30^\circ}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac{\frac {\sqrt 3} 2} {\frac 1 2}        | c = Cosine of $30^\circ$ and Sine of $30^\circ$ }} {{eqn | r = \sqrt 3       | c = multiplying top and bottom by $2$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 15^\circ       | r = \frac {\cos 15^\circ} {\sin 15^\circ}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac{\frac {\sqrt 6 + \sqrt 2} 4} {\frac {\sqrt 6 - \sqrt 2} 4}        | c = Cosine of 15 Degrees and Sine of 15 Degrees }} {{eqn | r = \frac {\sqrt 6 + \sqrt 2} {\sqrt 6 - \sqrt 2}       | c = simplifying }} {{eqn | r = \frac {\left({\sqrt 6 + \sqrt 2}\right)^2} {\left({\sqrt 6 - \sqrt 2}\right) \left({\sqrt 6 + \sqrt 2}\right)}       | c = multiplying top and bottom by $\sqrt 6 + \sqrt 2$  }} {{eqn | r = \frac {6 + 2 \sqrt 6 \sqrt 2 + 2 } {6 - 2}       | c = multiplying out, and Difference of Two Squares }} {{eqn | r = \frac {8 + 4 \sqrt 3} 4       | c = simplifying }} {{eqn | r = 2 + \sqrt 3       | c = dividing top and bottom by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 45 \degrees       | r = \frac {\cos 45 \degrees} {\sin 45 \degrees}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}        | c = Cosine of $45 \degrees$ and Sine of $45 \degrees$ }} {{eqn | r = 1       | c = multiplying top and bottom by $\dfrac {\sqrt 2} 2$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 60^\circ       | r = \frac {\cos 60^\circ} {\sin 60^\circ}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac{\frac 1 2} {\frac {\sqrt 3} 2}        | c = Cosine of $60^\circ$ and Sine of $60^\circ$ }} {{eqn | r = \frac {\sqrt 3} 3       | c = multiplying top and bottom by $2 \sqrt 3$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 75 \degrees       | r = \frac {\cos 75 \degrees} {\sin 75 \degrees}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac {\frac {\sqrt 6 - \sqrt 2} 4} {\frac {\sqrt 6 + \sqrt 2} 4}       | c = Cosine of $75 \degrees$ and Sine of $75 \degrees$ }} {{eqn | r = \frac {\sqrt 6 - \sqrt 2} {\sqrt 6 + \sqrt 2}       | c = simplifying }} {{eqn | r = \frac {\paren {\sqrt 6 - \sqrt 2}^2} {\paren {\sqrt 6 + \sqrt 2} \paren {\sqrt 6 - \sqrt 2} }       | c = multiplying top and bottom by $\sqrt 6 - \sqrt 2$  }} {{eqn | r = \frac {6 - 2 \sqrt 6 \sqrt 2 + 2} {6 - 2}       | c = multiplying out, and Difference of Two Squares }} {{eqn | r = \frac {8 - 4 \sqrt 3} 4       | c = simplifying }} {{eqn | r = 2 - \sqrt 3       | c = dividing top and bottom by $4$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 90^\circ       | r = \frac {\cos 90^\circ} {\sin 90^\circ}       | c = Cotangent is Cosine divided by Sine }} {{eqn | r = \frac 0 1       | c = Cosine of Right Angle and Sine of Right Angle }} {{eqn | r = 0       | c = }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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 = Tangent of $75 \degrees$ }} {{end-eqn}} {{qed}}
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}}
{{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 = Cotangent of $165 \degrees$ }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l = \cot 270^\circ       | r = \cot \left({360^\circ - 90^\circ}\right)       | c =  }} {{eqn | r = -\cot 90^\circ       | c = Cotangent of Conjugate Angle }} {{eqn | r = 0       | c = Cotangent of Right Angle }} {{end-eqn}} {{qed}}
{{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}}
{{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 = Cotangent of $60 \degrees$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cot 315^\circ       | r = \cot \left({360^\circ - 45^\circ}\right)       | c =  }} {{eqn | r = -\cot 45^\circ       | c = Cotangent of Conjugate Angle }} {{eqn | r = -1       | c = Cotangent of $45^\circ$ }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
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}}
{{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}}
{{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 = Cosine of $15 \degrees$ }} {{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}}
{{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 = Cosine of $30 \degrees$ }} {{eqn | r = \frac {2 \sqrt 3} 3       | c = multiplying top and bottom by $2 \sqrt 3$ }} {{end-eqn}} {{qed}}
{{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}}
{{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}}