How do you use the double angle or half angle formulas to simplify #cos^2 (5B) - sin^2 (5B)#?

1 Answer
Oct 17, 2015

Simplify: #cos^2 (5B) - sin ^2 (5B)#

Ans: #sin (10B + pi/2)#

Explanation:

Call arc (5B) = x
#y = cos^2 x - sin^2 x = (cos x - sin x)(cos x + sin x)#
Apply the trig identities:
#cos x - sin x = sqrt2cos (x + pi/4)#
#cos x + sin x = sqrt2sin (x + pi/4)#
#y = [sqrt2.cos (x + pi/4)][sqrt2.sin (x + pi/4)] =#
#y = 2sin (x + pi/4).cos (x + pi/4) = sin 2(x + pi/4) = sin (2x + pi/2)#
Replace back x = 5B
#y = sin (10B + pi/2)#