Question

Prove that, $\sin x(1+\tan x)+\cos x(1+\cot x)=\sec x+cosecx$

Answer 1

$LHS=\sin x(1+\tan x)+\cos x(1+\cot x)=\sec x+cosecx$

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

$=(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x})(\sin x+\cos x)$

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

$=(\frac{\sin x}{\sin x\cos x}+\frac{\cos x}{\sin x\cos x})$ .......... $[sin²\theta +cos²\theta =1]$

$=\frac{1}{\cos x}+\frac{1}{\sin x}=\sec x+cosecx$

$=RHS$