Question

Prove the following identities (1-16)
$\mathrm{cosec}x\left(\sec x-1\right)-\cot x\left(1-\cos x\right)=\tan x-\sin x$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\mathrm{cosec}x\left(\sec x-1\right)-\cot x\left(1-\cos x\right) \\ =\frac{1}{\sin x}\left(\frac{1}{\cos x}-1\right)-\frac{\cos x}{\sin x}\left(1-\cos x\right) \\ =\frac{1}{\sin x}\left(\frac{1-\cos x}{\cos x}\right)-\frac{\cos x}{\sin x}\left(1-\cos x\right) \\ =\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{1}{\cos x}-\cos x\right) \\ =\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{1-{\cos }^{2}x}{\cos x}\right) \\ =\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{{\sin }^{2}x}{\cos x}\right) \\ =\left(1-\cos x\right)\left(\frac{\sin x}{\cos x}\right) \\ =\frac{\sin x}{\cos x}-\sin x \\ =\tan x-\sin x \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$