Question

Why can `cos^4 θ` be written as `(cos^2 θ)^2` ?

Why can the θ have a square after that?

Answer 1

As proved below.

Explanation:

As per theory of indices,

`a^m * a^n = a^(m+n)`

`(a^m)^n = a^(mn)`

`cos^4 theta = cos^(2 + 2) theta`

`=> cos^2 theta * cos^2 theta = (cos^2 theta)^2`

Answer 2

`cos^2 theta` is shorthand for `(cos theta)^2` and `cos^4 theta` is shorthand for `(cos theta)^4.` So,

`cos^4 theta = (cos theta)^4 = ( (cos theta)^2) ^2 = (cos^2 theta)^2`