prove-that-cos-6-a-sin-6-a-cos-2a-1-1-4-sin-2-2a-

Question Number 104894 by bramlex last updated on 24/Jul/20

p r o v e t h a t \cos^{6} a - \sin^{6} a =

\cos 2 a (1 - \frac{1}{4} \sin^{2} 2 a)

Answered by john santu last updated on 24/Jul/20

(1) \cos^{6} a - \sin^{6} a = (\cos^{2} a - \sin^{2} a)

(\cos^{4} a + \sin^{2} a \cos^{2} a + \sin^{4} a)

= (\cos^{2} a - \sin^{2} a) {{(\cos^{2} a + \sin^{2} a)}^{2} - \frac{1}{4} {(2 \sin a \cos a)}^{2}}

= \cos 2 a {1 - \frac{1}{4} \sin 2 a} (J S ⊛)