Question

Prove: $\sin A(1+\tan A)+\cos A(1+\cot A)=\sec A+co\sec A$

Answer 1

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

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

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

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

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

$=RHS$

(Here, $x=A$)