Question

Simplifying Trigonometric Expressions-

Simplify the trigonometric expression$\frac{1}{1-\sin \alpha }+\frac{1}{1+\sin \alpha }$

Short answer

On simplifying expression we get$2{\sec }^2\alpha$

Step-by-step solution

Step 1. Given information.

A trigonometric expression as$\frac{1}{1-\sin \alpha }+\frac{1}{1+\sin \alpha }$

Step 2. Concept used.

First simplify the given expression into the simplest form as possible. Do it with the help of trigonometric identities.

Step 3. Calculation.

Now, simplify$\frac{1}{1-\sin \alpha }+\frac{1}{1+\sin \alpha }$ using identities:

$\Rightarrow \frac{1+\sin \alpha +1-\sin \alpha }{(1-\sin \alpha )(1+\sin \alpha )}$

$$\begin{gathered} \Rightarrow \frac{2}{1-{\sin }^2\alpha } \\ \begin{array}{l}\Rightarrow \frac{2}{{\cos }^2\alpha }\left(\because 1-{\sin }^2\alpha ={\cos }^2\alpha \right)\\ \Rightarrow 2{\sec }^2\alpha \left(\because \frac{1}{{\cos }^2\alpha }={\sec }^2\alpha \right)\end{array} \end{gathered}$$