Question

How do I solve this trigonometric function?

Answer 1

Answer:
`(1)/(cos \theta)=sec \theta`

Explanation:

We have the following expression:

`(1)/(2)((1+sin \theta)/(cos \theta)+(cos \theta)/(1+sin \theta))`

Firstly, we have to solve what is inside the parenthesis. Let's begin by calculating the least common multiple (l.c.m), which is
`cos \theta(1+sin \theta)`:

`(1)/(2)(((1+sin \theta)^(2) +cos^(2) \theta)/(cos \theta(1+sin \theta)))`

`(1)/(2)((1+2sin \theta+sin^(2) \theta+cos^(2) \theta)/(cos \theta(1+sin \theta)))`

Since
`sin^(2) \theta+cos^(2) \theta=1`:

`(1)/(2)((1+2sin \theta+1)/(cos \theta(1+sin \theta)))`

`(1)/(2)((2(1+sin \theta))/(cos \theta(1+sin \theta)))`

Simplifying:

`(1)/(cos \theta)=sec \theta`