Question

Find $x$ from the following equation:-
$\mathrm{cosec}\left(90+\theta \right)+x\cos \theta \cot \left(90+\theta \right)=\sin \left(90+\theta \right)$

Answer 1

Given equation is,

$\csc (90+\theta )+x\cos \theta \cot (90+\theta )=\sin (90+\theta )$

We know that ,

$cosec(90+\theta )=\sec \theta$

$\cot (90+\theta )=-\tan \theta$

$\sin (90+\theta )=\cos \theta$

Put these values in the given equation,we get

$\Rightarrow \sec \theta +x\cos \theta (-\tan \theta )=\cos \theta$

$\Rightarrow \sec \theta +x\cos \theta (-\frac{\sin \theta }{\cos \theta })=\cos \theta$

$\Rightarrow \sec \theta +x(-\sin \theta )=\cos \theta$

$\Rightarrow x(-\sin \theta )=\cos \theta -\sec \theta$

$\Rightarrow x(-\sin \theta )=\cos \theta -\frac{1}{\cos \theta }$

$\Rightarrow x(-\sin \theta )=\frac{{\cos }^2\theta -1}{\cos \theta }$

$\Rightarrow x(-\sin \theta )=-\frac{{\sin }^2\theta }{\cos \theta }$

$\Rightarrow x=\frac{\sin \theta }{\cos \theta }$

$\Rightarrow x=\tan \theta$

Hence, $x=\tan \theta$