Question

Prove the following identity :

secA(1 + sinA)(secA - tanA) = 1

Answer 1

LHS = secA(1 + sinA)(secA - tanA) 

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

= ${\displaystyle \frac{(1+\sin A)}{\cos A}\left(\frac{1-\sin A}{\cos A}\right)=\frac{1-{\sin }^{2}A}{{\cos }^{2}A}}$

= ${\displaystyle \left(\frac{{\cos }^{2}A}{{\cos }^{2}A}\right)=1}$ = RHS