Question

Prove that : (cosecA - sin A) (secA-cosA) =1/tan A +cot A

Answer 1

These are the steps of this question

Answer 2

Answer and Explanation:

To prove : $(\csc A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$

Proof :

Taking LHS,

$=(\csc A-\sin A)(\sec A-\cos A)$

$=(\frac{1}{\sin A}-\sin A)(\frac{1}{cos A}-\cos A)$

$=(\frac{1-\sin^2A}{\sin A})(\frac{1-\cos^2A}{cos A})$

$=(\frac{\cos^2A}{\sin A})(\frac{\sin^2A}{cos A})$

$LHS=\sin A \cos A$

Taking RHS,

$=\frac{1}{\tan A+\cot A}$

$=\frac{1}{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}$

$=\frac{1}{\frac{\sin^2A+\cos^2A}{\sin A\cos A}}$

$=\frac{1}{\frac{1}{\sin A\cos A}}$

$RHS=\sin A\cos A$

LHS=RHS

Hence proved.