Question

Question #bdbc2

Answer 1

See below.

Explanation:

To prove that: `cos^4(x) = 3/8+1/2cos(2x)+1/8cos(4x)`

First we have that:

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

We can now make use of trig identity:

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

So we can say that:

`cos^4(x) = (1/2cos(2x)+1/2)(1/2cos(2x)+1/2)`

Expanding these brackets gives us:

`1/4cos^2(2x)+1/2cos(2x)+1/4`

Now using the identity again we can say that

`cos^2(2x) = 1/2cos(4x) +1/4`

Substituting this into our expression gives:

`cos^4(x) = 1/4(1/2cos(4x)+1/2)+1/2cos(2x)+1/4)`

`=1/8cos(4x)+1/8 +1/2cos(2x) +1/4`

`=1/8cos(4x)+1/2cos(2x)+3/8`

As required.