Question

Prove:$\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2\sec A$

Answer 1

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

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

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

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

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

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

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

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

$=2\sec A$

R.H.S.