Question

$Simplify\left(i\right)\frac{cot(90+\theta )sec(-\theta )\tan (180\theta -\theta )}{sec(360-\theta )\sin (180+\theta )cot(90-\theta )}$

Answer 1

Dear Student,
$$\begin{gathered} \frac{cot\left(90°-\theta \right)sec\left(-\theta \right)\tan \left(180°-\theta \right)}{sec\left(360°-\theta \right)\sin \left(180°-\theta \right)cot\left(90°-\theta \right)} \\ =\frac{\tan {\displaystyle \theta }{\displaystyle s}{\displaystyle e}{\displaystyle c}{\displaystyle \theta }{\displaystyle \left(-\tan \theta \right)}}{sec\theta \sin \theta \tan \theta } \\ =\frac{-\tan \theta }{\sin \theta } \\ =\frac{{\displaystyle \frac{-\sin \theta }{\cos \theta }}}{\sin \theta }\left[\tan \theta =\frac{\sin \theta }{\cos \theta }\right] \\ =\frac{-1}{\cos \theta } \\ =\mathbf{-}\mathit{s}\mathit{e}\mathit{c}\mathit{\theta } \end{gathered}$$
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