Mathematics - Trigonometric Equation Question with Solution | TestHub

Given:

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

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

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

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

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

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

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

So,

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

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

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