Prove the following identities (1-16)


Prove the following identities (1-16)

$\operatorname{cosec} x(\sec x-1)-\cot x(1-\cos x)=\tan x-\sin x$


$\mathrm{LHS}=\operatorname{cosec} x(\sec x-1)-\cot x(1-\cos x)$

$=\frac{1}{\sin x}\left(\frac{1}{\cos x}-1\right)-\frac{\cos x}{\sin x}(1-\cos x)$

$=\frac{1}{\sin x}\left(\frac{1-\cos x}{\cos x}\right)-\frac{\cos x}{\sin x}(1-\cos x)$

$=\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{1}{\cos x}-\cos x\right)$

$=\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{1-\cos ^{2} x}{\cos x}\right)$

$=\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{\sin ^{2} x}{\cos x}\right)$

$=(1-\cos x)\left(\frac{\sin x}{\cos x}\right)$

$=\frac{\sin x}{\cos x}-\sin x$

$=\tan x-\sin x$


Hence proved.

Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now