Question

Prove the following:
$cos6x=32co{s}^{6}x-48co{s}^{4}x+18co{s}^{2}x-1$

Answer 1

Use $\cos 3x=4{\cos }^{3}x-3\cos x$

$LHS=\cos 6x=\cos 3\left(2x\right)=4{\cos }^{3}2x-3\cos 2x$

$=4{(2{\cos }^{2}x-1)}^3-3(2{\cos }^{2}x-1)$
$=4(8{\cos }^{6}x-1-12{\cos }^{4}x+6{\cos }^{2}x)-6{\cos }^{2}x+3$
$=32co{s}^{6}x-48co{s}^{4}x+18co{s}^{2}x-1=R.H.S$

$\because L.H.S=R.H.S$

$\therefore \cos 6x=32co{s}^{6}x-48co{s}^{4}x+18co{s}^{2}x-1$