Question

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

Answer 1

Given expression is
`(1)/(1+ sin \theta) + (1)/(1 - sin \theta)`

Now we can combine it as

`(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`


`(2)/(1 - sin^2 \theta)`

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


`(2)/(1 - sin^2 \theta) = (2)/(cos^2 \theta)`

As we know
`(1)/(cos \theta) = sec \theta`

So
`(2)/(cos^2 \theta) = 2 sec^2 \theta`

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