The set of all values of λ for which the equation ^ 2 2x - 2 ^ 4 x...

Given:

$\lambda = \cos^{2}2x - 2\sin^{4}x - 2\cos^{2}x$

$⇒ \lambda = {\left(2\cos^{2}x - 1\right)}^{2} - 2{\left(1 - \cos^{2}x\right)}^{2} - 2\cos^{2}x$

$⇒ \lambda = 4\cos^{4}x - 4\cos^{2}x + 1 - 2\left(1 - 2\cos^{2}x + \cos^{4}x\right) - 2\cos^{2}x$

$⇒ \lambda = 2\cos^{4}x - 2\cos^{2}x + 1 - 2$

$⇒ \lambda = 2\cos^{4}x - 2\cos^{2}x - 1$

$⇒ \lambda = 2\left[\cos^{4}x - \cos^{2}x - \dfrac{1}{2}\right]$

$⇒ \lambda = 2\left[{\left(\cos^{2}x - \dfrac{1}{2}\right)}^{2} - \dfrac{3}{4}\right]$

So,

{\lambda}_\max = 2\left(\dfrac{1}{4} - \dfrac{3}{4}\right) = - 1 (max Value)

{\lambda}_\min = 2\left(0 - \dfrac{3}{4}\right) = - \dfrac{3}{2} (Minimum Value)

So, range of $\lambda$ is\left[- \dfrac{3}{2}, - 1\right]$