Question

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

Answer 1

Simplifying the LHS of $\frac{1+\sin A}{\cos A}+\frac{\cos A}{1+\sin A}=2\sec A$.

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

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

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

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

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

$=\frac{2}{\cos A}$

$=2\sec A$

$=RHS$