Question

Question #71188

Answer 1

`1-sin x`

Explanation:

`(1 - sin^2 x)/(1+sin x)`

We can use the Pythagorean identity `color(blue)(sin^2x + cos^2x = 1)` and rearrange it as `cos^2x = 1 - sin^2 x`.

`cos^2x / (1+sin x)`

Now we can multiply the numerator and denominator by `(1-sin x)/(1-sin x)`, which is a valid step since it equals `1`.

`cos^2x / (1+sin x) * color(blue)((1-sin x)/(1-sin x))`

`(cos^2x (1-sin x)) / ((1+sin x)(1-sin x))`

Since `(a+b)(a-b) = a^2 - b^2`, we can rewrite the denominator as

`(cos^2x (1-sin x)) / (1^2 - sin^2x)`

As stated above, `1 - sin^2x = cos^2x`.

`(cancel(cos^2x) (1-sin x)) / cancel(cos^2x)`

Finally, we can cancel out `cos^2x`, leaving us with `color(red)(1-sin x)`.