Question

Prove that the identities:
$\cos 5\theta =16{\cos }^6\theta -20{\cos }^3\theta +5\cos \theta$

Answer 1

$\cos 5\theta =\cos (2\theta +3\theta )$

$=\cos 2\theta \cos 3\theta -\sin 2\theta \sin 3\theta$

$=(2{\cos }^2\theta -1)(4{\cos }^3\theta -3\cos \theta )-\left(2\sin \theta \cos \theta \right)(3\sin \theta -4{\sin }^3\theta )$

$=8{\cos }^5\theta -6{\cos }^3\theta -4{\cos }^3\theta +3\cos \theta -6{\sin }^2\theta \cos \theta +8{\sin }^4\theta \cos \theta$

$=8{\cos }^5\theta -10{\cos }^3\theta +3\cos \theta -6{\sin }^2\theta \cos \theta +8{\sin }^4\theta \cos \theta$

$=8{\cos }^5\theta -10{\cos }^3\theta +3\cos \theta -6\cos \theta (1-{\cos }^2\theta )+8{(1-{\cos }^2\theta )}^2\cos \theta$

$=8{\cos }^5\theta -10{\cos }^3\theta +3\cos \theta -6\cos \theta +6{\cos }^3\theta +8\cos \theta (1-2{\cos }^2\theta +{\cos }^4\theta )$

$=8{\cos }^5\theta -4{\cos }^3\theta -3\cos \theta +8\cos \theta -16{\cos }^3\theta +8{\cos }^5\theta$

$=16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

$\therefore \cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

Hence proved.