Question

If $si{n}^6\theta +co{s}^6\theta -1=\lambda si{n}^2\theta co{s}^2\theta ,$ find the value of $\lambda$.

Answer 1

$\Rightarrow \sin {}^6\theta +\cos {}^6\theta -1=\lambda \sin {}^2\theta \cos {}^2\theta$

$\Rightarrow {(\sin {}^2\theta +\cos {}^2\theta )}^3-3\sin {}^2\theta \cos {}^2\theta (\sin {}^2\theta +\cos {}^2\theta )=\lambda \sin {}^2\theta \cos {}^2\theta$
$\Rightarrow 1-3\sin {}^2\theta \cos {}^2\theta -1=\lambda \sin {}^2\theta \cos {}^2\theta$
$\therefore \lambda =-3$