trigonometry
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Verify that following are identities:
1. cos 3t = 4 cos³ t3 cos t
2. sin 4x = 8 sin x cos³ x4 x cos x
(use a doubleangle identity)

Reiny
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 doubleangle formulas for each one
simplify very carefully
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