Question 901559
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You need the following facts:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sec\varphi\ =\ \frac{1}{\cos\varphi}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \csc\varphi\ =\ \frac{1}{\sin\varphi}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan\varphi\ =\ \frac{1}{\cot\varphi}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan\varphi\ =\ \frac{\sin\varphi}{\cos\varphi}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\varphi\ =\ 1\ -\ \sin^2\varphi]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{3\pi}{2}\ <\ \theta\ <\ 2\pi\ \Right\ \theta\ \in\ QIV]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \varphi\ \in\ QIV\ \Right\ \sin\varphi\ <\ 0]


Given


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sec\theta\ =\ \frac{13}{5}]


then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos\theta\ =\ \frac{5}{13}]


then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\theta\ =\ \frac{25}{169}]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin^2\theta\ =\ \frac{144}{169}]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin\theta\ =\ -\frac{12}{13}]


Minus because *[tex \LARGE \theta\ \in\ QIV]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \csc\theta\ =\ -\frac{13}{12}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan\theta\ =\ \frac{-\frac{12}{13}}{\frac{5}{13}}\ =\ -\frac{12}{5}]


and finally


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cot\theta\ =\ -\frac{5}{12}]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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