Question

please help!!! Simplify the trigonometric expression. Show your work.

Answer 1

Given expression is $\frac{1}{1+ sin \theta} + \frac{1}{1 - sin \theta}$

Now we can combine it as
$\frac{1 - sin \theta + 1 + sin \theta}{(1+sin \theta)(1 - sin \theta)}$

Now we can use identity as $(a+b)(a-b) = a^2 - b^2$

$\frac{2}{1 - sin^2 \theta}$

Now we can use identity as $1 - sin^2 \theta = cos^2 \theta$

$\frac{2}{1 - sin^2 \theta} = \frac{2}{cos^2 \theta}$

As we know $\frac{1}{cos \theta} = sec \theta$

So $\frac{2}{cos^2 \theta} = 2 sec^2 \theta$

So given expression equal to $2sec^2 \theta$