Question

Simplify
1/1+sin theta + 1/1-sin theta

Answer 1

\[\frac{1}{1+sin~\theta} + \frac{1}{1-sin~\theta}\]
Multiply and divide both terms by their conjugate [To make the denominators same]

\[\frac{1 * (1-sin\theta)}{(1+sin~\theta)(1-sin\theta)} + \frac{1 * (1+sin\theta)}{(1-sin~\theta)(1+sin\theta)}\]
Now you've got the denominators to be equal!

Then use these rules: \(1-sin^2\theta = cos^2\theta ~~~AND~~~\frac{1}{cos\theta} = sec\theta\)

After simplifying more you'll get:

\[\frac{1-sin\theta}{1-sin^2~\theta} + \frac{1+sin\theta}{1-sin^2~\theta} = \frac{2}{cos^2\theta} = 2sec^2\theta\]