Question

$\cos 5\theta =0$ if $64{\cos }^4\theta -80{\cos }^2\theta +1625$ is equal to

Answer 1

Given $\cos 5\theta =0$
$\cos 4\theta \cos \theta -\sin 4\theta \sin \theta =0$
$(2{\cos }^{2}2\theta -1)\cos \theta -2\sin 2\theta \cos 2\theta \sin \theta =0$
$\cos \theta [2{\cos }^{2}2\theta -1-4{\sin }^2\theta \cos 2\theta ]=0$
$\cos \theta [4{\cos }^{2}2\theta -2\cos 2\theta -1]=0$
$\Rightarrow \cos \theta =0$ or $4{\cos }^{2}2\theta -2\cos 2\theta -1=0$
$\Rightarrow \cos \theta =0$ or $\cos 2\theta =\frac{1\pm \sqrt{5}}{4}$
$\Rightarrow {\cos }^2\theta =\frac{5\pm \sqrt{5}}{8}$
$\Rightarrow (8{\cos }^2\theta -5{)}^2=5$
$\Rightarrow 64{\cos }^4\theta -80{\cos }^2\theta +20=0$
$\Rightarrow 64{\cos }^4\theta -80{\cos }^2\theta +1625=1605$