Question

If $\sin 2\theta =\frac{1}{4}$
Find ${\cos }^6\theta +{\sin }^6\theta$

Answer 1

$\sin 2\theta =\frac{1}{4}2\cos \theta \sin \theta =\frac{1}{4}\cos \theta \sin \theta =\frac{1}{8}\therefore \sin {}^6\theta +\cos {}^6\theta ={(\sin {}^2\theta +\cos {}^2\theta )}^3+3(\sin {}^2\theta +\cos {}^2\theta )\left(\sin {}^2\theta \cos {}^2\theta \right)={\left(1\right)}^3+3\left(1\right)\frac{1}{64}=1+\frac{3}{64}=\frac{67}{64}$