Question

Verify that following are identities:


1. cos 3t = 4 cos³ t-3 cos t

2. sin 4x = 8 sin x cos³ x-4 x cos x

(use a double-angle identity)

Answer 1

1.
LS = cos 3t
= cos(2t + t)
= cos2tcost - sin2tsint
= (2cos^2 t - 1)(cost) - 2sintcostsint
= 2cos^3 t - cost - 2sin^2 t cost
= 2cos^3 t - cost - 2(1 - cos^2 t)cost
= 2cos^3 t - cost - 2cost + 2cos^3 t
= 4cos^3 t - 3cost
= RS

for #2, start with
LS = sin(2x + 2x)
= sin2xcos2x + cos2x sin2x
then use the double-angle formulas for each one
simplify very carefully