Question #bdbc2

1 Answer
May 23, 2016

Answer:

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.