Question

Prove that:

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

Answer 1

To prove:

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

Taking LHS

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

$=\frac{\left(\cos A\right)\left(\cos A\right)+(1+\sin A)(1+\sin A)}{(1+\sin A)\left(\cos A\right)}$ ( Taking LCM of denominator)

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

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

$$\begin{gathered} =\frac{2(1+\sin A)}{(1+\sin A)\cos A} \\ =\frac{2}{\cos A} \\ =2SecA \end{gathered}$$

$=RHS$

Hence, proved $\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2SecA$