Using the double angle of half angle formula, how do you simplify #cos^2 5theta- sin^2 5theta#?

1 Answer
Apr 25, 2015

There is another simple way to simplify this.
#cos^2 5x - sin^2 5x = (cos 5x - sin 5x)(cos 5x + sin 5x)#
Use the identities:
#cos a - sin a = -(sqrt2) *(sin (a - Pi/4))#

#cos a + sin a = (sqrt2)* (sin (a + Pi/4))#

So this becomes:
#-2 * sin (5x - Pi/4) * sin (5x + Pi/4) #.

Since #sin a * sin b = 1/2 (cos(a-b)-cos(a+b))#, this equation can be rephrased as (removing the parentheses inside the cosine):

#-(cos(5x - Pi/4-5x-Pi/4)-cos(5x - Pi/4+5x + Pi/4)) #

This simplifies to:

#-(cos(-pi/2)-cos(10x))#

The cosine of #-pi/2# is 0, so this becomes:

#-(-cos(10x))#

#cos(10x)#

Unless my math is wrong, this is the simplified answer.