Question

Prove that $\frac{1}{\left(1+\sin \theta \right)}+\frac{1}{\left(1-\sin \theta \right)}=2{\sec }^2\theta$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{1}{\left(1+\sin \theta \right)}+\frac{1}{\left(1-\sin \theta \right)} \\ =\frac{\left(1-\sin \theta \right)+\left(1+\sin \theta \right)}{\left(1+\sin \theta \right)\left(1-\sin \theta \right)} \\ =\frac{2}{1-{\sin }^2\theta } \\ =\frac{2}{{\cos }^2\theta } \\ =2{\sec }^2\theta \\ =\mathrm{RHS} \end{gathered}$$