Question

Simplify

$\frac{\sin ({270}^o+x){\cos }^3({720}^o-x)-\sin ({270}^o-x){\sin }^3({540}^o+x)}{\sin ({90}^o+x)\sin (-x)-{\cos }^2({180}^o-x)}+\frac{cot({270}^o-x)}{cose{c}^2({450}^o+x)}$.

Answer 1

$\frac{\sin ({270}^o+x){\cos }^3({720}^o-x)-\sin ({270}^o-x){\sin }^3({540}^o+x)}{\sin ({90}^o+x)\sin (-x)-{\cos }^2({180}^o-x)}+\frac{\cot ({270}^o-x)}{cose{c}^2({450}^o+x)}$

We know, $\sin ({270}^o+\theta )=-\cos \theta ,$ $\sin ({270}^o-\theta )=\cos \theta$

$\cos ({720}^o-\theta )=\cos \theta ,$ $\sin ({540}^o+\theta )=\cos \theta$

$\cot ({270}^o-\theta )=\tan \theta$ and $cosec({450}^o+\theta )=\sec \theta$

$\Rightarrow$ $\frac{-\cos x\times {\cos }^{3}x-(\cos x\times {\cos }^{3}x)}{-\cos x\sin x-{\cos }^{2}x}+\frac{\tan x}{{\sec }^{2}x}$

$\Rightarrow$ $\frac{-{\cos }^{4}x-{\cos }^{4}x}{-\cos x\sin x-{\cos }^{2}x}+\frac{\sin x}{\cos x}\times {\cos }^{2}x$

$\Rightarrow$ $\frac{-({\cos }^{4}x+{\cos }^{4}x)}{-\cos x(\sin x+\cos x)}+\sin x\cos x$

$\Rightarrow$ $\frac{2{\cos }^{4}x}{\cos x(\sin x+\cos x)}+\sin x\cos x$

$\Rightarrow$ $\frac{2{\cos }^{3}x}{\sin x+\cos x}+\sin x\cos x$

$\Rightarrow$ $\frac{2{\cos }^{3}x+{\sin }^{2}x\cos x+\sin x{\cos }^{2}x}{\sin x+\cos x}$