Question

If $f\left(x\right)=3[{\sin }^4(\frac{3\pi }{4}-x)+{\sin }^4(3\pi +x\left)\right]-2[{\sin }^6(\frac{\pi }{2}+x)+{\sin }^6(5\pi -x\left)\right]$ then, for all permissible values of $x$, $f\left(x\right)$ is -

• A. $-1$
• B. $0$
• C. $1$
• D. Not a constant function.

Answer 1

The correct option is D Not a constant function.

$3[{\sin }^4(\frac{3\pi }{4}-x)+{\sin }^4(3\pi +x)]-2[{\sin }^6(\frac{\pi }{2}+x)+{\sin }^6(5\pi -x)]$

$\Rightarrow 3[{(\frac{\cos x}{\sqrt{2}}-\frac{\sin x}{\sqrt{2}})}^2+{\sin }^{4}x]-2[{\cos }^{6}x+{\sin }^{6}x]$

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

$\Rightarrow 3[\frac{1}{2}-\frac{\sin 2x}{2}+\frac{1}{4}-\frac{\cos 2x}{2}+\frac{{\cos }^{2}2x}{4}]-2[1-\frac{3{\sin }^{2}2x}{4}]$

$\Rightarrow (\frac{3}{2}+\frac{3}{4}-2)-\frac{3}{2}(\cos 2x+\sin 2x)+\frac{3}{4}({\cos }^{2}2x+{\sin }^{2}2x)+\frac{3{\sin }^{2}2x}{4}$

$\Rightarrow 1+\frac{3}{4}\left({\sin }^{2}2x-(\sin 2x+\cos 2x)\right)$

As it cannot be made into a constant function.

Option D is the answer.