hcju/proofwikiqa huggingface dataset:
{{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 = Cosine of $75 \degrees$ }} {{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}}
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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l=\csc \left({\pi - \theta}\right)       | r=\frac 1 {\sin \left({\pi - \theta}\right)}       | 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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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 = Secant of $45 \degrees$ }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
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}}
{{begin-eqn}} {{eqn | l = \csc 15^\circ       | r = \frac 1 {\sin 15^\circ}       | c = Cosecant is Reciprocal of Sine }} {{eqn | r = \frac 1 {\frac {\sqrt 6 - \sqrt 2} 4}       | c = Sine of 15 Degrees }} {{eqn | r = \frac 4 {\sqrt 6 - \sqrt 2}       | c = multiplying top and bottom by $4$ }} {{eqn | r = \frac {4 \left({\sqrt 6 + \sqrt 2}\right)} {\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 {4 \left({\sqrt 6 + \sqrt 2}\right)} {6 - 2}       | c = Difference of Two Squares }} {{eqn | r = \sqrt 6 + \sqrt 2       | c = simplifying }} {{end-eqn}} {{qed}}
{{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}}
{{begin-eqn}} {{eqn | l = \csc 45^\circ       | r = \frac 1 {\sin 45^\circ}       | c = Cosecant is Reciprocal of Sine }} {{eqn | r = \frac 1 {\frac {\sqrt 2} 2}       | c = Sine of 45 Degrees }} {{eqn | r = \sqrt 2       | c = multiplying top and bottom by $2 \sqrt 2$ }} {{end-eqn}} {{qed}}
{{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}}
{{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 = Sine of $75 \degrees$ }} {{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}}
{{begin-eqn}} {{eqn | l = \csc 90^\circ       | r = \frac 1 {\sin 90^\circ}       | c = Cosecant is Reciprocal of Sine }} {{eqn | r = \frac 1 1       | c = Sine of 90 Degrees }} {{eqn | r = 1       | c =  }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
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}}
{{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}}
{{begin-eqn}} {{eqn | l = \csc 210^\circ       | r = \csc \left({360^\circ - 150^\circ}\right)       | c =  }} {{eqn | r = -\csc 150^\circ       | c = Cosecant of Conjugate Angle }} {{eqn | r = -2       | c = Cosecant of $150^\circ$ }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l = \csc 255^\circ       | r = \csc \left({360^\circ - 105^\circ}\right)       | c =  }} {{eqn | r = -\csc 105^\circ       | c = Cosecant of Conjugate Angle }} {{eqn | r = -\left({\sqrt 6 - \sqrt 2}\right)       | c = Cosecant of $105^\circ$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \csc 270^\circ       | r = \csc \left({360^\circ - 90^\circ}\right)       | c =  }} {{eqn | r = -\csc 90^\circ       | c = Cosecant of Conjugate Angle }} {{eqn | r = -1       | c = Cosecant of Right Angle }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \csc 285^\circ       | r = \csc \left({360^\circ - 75^\circ}\right)       | c =  }} {{eqn | r = -\csc 75^\circ       | c = Cosecant of Conjugate Angle }} {{eqn | r = -\left({\sqrt 6 - \sqrt 2}\right)       | c = Cosecant of 75 Degrees }} {{end-eqn}} {{qed}}
{{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}}
{{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}}
{{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 = Cosecant of $30 \degrees$ }} {{end-eqn}} {{qed}}
{{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}}
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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{begin-eqn}} {{eqn | l = \cot \left({a + b}\right)       | r = \frac{\cos \left({a + b}\right)} {\sin \left({a + b}\right)}        | 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 the 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}}
=== 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}} === Sufficient Condition === Let $k$ be an odd integer. Then by Odd Integer 2n + 1: :$k = 2 n + 1$ where $n \in \Z$. Then: {{begin-eqn}} {{eqn | l = \frac k 2       | r = \frac {2 n + 1} 2       | c =  }} {{eqn | r = \frac {2 n} 2 + \frac 1 2       | c =  }} {{eqn | r = n + \frac 1 2       | c =  }} {{end-eqn}} thus showing that $\dfrac k 2$ is a half-integer. {{qed}} Category:Integers Category:Odd Integers Category:Numbers 5mfab4beguz5qevypjz3d2aboc2jh8x
{{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}}
{{begin-eqn}} {{eqn | l = \cos \paren {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}}
{{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}}
{{begin-eqn}} {{eqn | l = \map \cos {x + \pi}       | r = \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }       | c = Cosine Exponential Formulation }} {{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 = Cosine Exponential Formulation }} {{end-eqn}} {{qed}}
{{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}}
{{begin-eqn}} {{eqn | l = \cos \paren {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}}
{{begin-eqn}} {{eqn | l = \tan \left({x + 2 \pi}\right)       | r = \frac {\sin \left({x + 2 \pi}\right)} {\cos \left({x + 2 \pi}\right)}        | 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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}
{{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}}