Question

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

Answer 1

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)`