Question

I need help simplifying this expression?: cos^2 (x) /(1+sin(x))

Answer 1

Recall that `cos^2x = 1- sin^2x`.

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

`=((1 - sinx)(1 + sinx))/(sinx +1)`

`=1 - sinx`

Hopefully this helps!

Answer 2

`1-sinx`

Explanation:

`"using the "color(blue)"trigonometric identity"`

`•color(white)(x)sin^2x+cos^2x=1`

`rArrcos^2x=1-sin^2x`

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

`1-sin^2x" is a "color(blue)"difference of squares"`

`[•color(white)(x)a^2-b^2=(a-b)(a+b)]`

`=((1-sinx)cancel((1+sinx)))/cancel((1+sinx))=1-sinx`

Answer 3

The answer is `1-sinx`.

Explanation:

Simplifying `cos^2x/(1+sinx`:

Because of the identity `sin^2x+cos^2x=1` (based on the Pythagorean Theorem), you can move the `sin^2x` to the other side of the equation and you get
`cos^2x=1-sin^2x`.

Substitute it in the equation, and you will get `(1-sin^2x)/(1+sinx`.

You can think of `1-sin^2x` as `1^2 - (sinx)^2`.

Based on the rule `(x-y)(x+y)=x^2-y^2`, you can therefore write `1-sin^2x` as `(1-sinx)(1+sinx)`.

Substitute it into the equation and you get `((1-sinx)(1+sinx))/(1+sinx`.

However, in both the numerator and denominator there is a `1+sinx`. These can cancel out, and, thus, you will get `1-sinx` as your final answer.

Hope this helps!