Question

How do you evaluate `cos^2 (-135)`?

Answer 1

`cos^2(-135^o)=1/2`

Explanation:

First of all, we should assume that `-135` is degrees, not radians.

Secondly, recall the definition of a function cosine.
Cosine of an angle is an abscissa (X-coordinate) of the point on a unit circle at the end of a radius that makes this angle in the counterclockwise direction from the positive direction of X-axis.

From this definition and, as seen from the picture, it is obvious that `cos(x)=cos(-x)` and
`cos(180^o-x)=-cos(x)`

Let's now find the value of `cos(-135^o)`.
From `cos(-x)=cos(x)` follows that
`cos(-135^o)=cos(135^o)`

From `cos(180^o-x)=-cos(x)` follows that
`cos(135^o)=cos(180^o-45^o)=-cos(45^o)=-sqrt(2)/2`

Hence, `cos^2(-135^o)=(-sqrt(2)/2)^2 = 1/2`