Question

Prove that $\frac{\cos A\cot A-\sin A\tan A}{cosecA-\sec A}=1+\sin A\cos A$

Answer 1

$LHS=\frac{\cos A\cot A-\sin A\tan A}{\csc A-\sec A}=\frac{\frac{\cos {}^{2}A}{\sin A}-\frac{\sin {}^{2}A}{\cos A}}{\frac{1}{\sin A}-\frac{1}{\cos A}}=\frac{\cos {}^{3}A-\sin {}^{3}A}{\cos A-\sin A}=\frac{(\cos A-\sin A)(\cos {}^{2}A+\cos A\sin A+\sin {}^{2}A)}{(\cos A-\sin A)}=1+\cos A\sin A=RHS$