Question

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

• A. $-1$
• B. $0$
• C. $1$
• D. $noaconstantfunction$

Answer 1

The correct option is C $1$

Given

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

$(-\cos x{)}^4+(-\sin x{)}^4$

${\sin }^{4}x+{\cos }^{4}x$

$({\sin }^{2}x+{\cos }^{2}x{)}^2-2{\sin }^{2}x{\cos }^{2}x$

$1-2{\sin }^{2}x{\cos }^2$

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

${\cos }^{6}x-{\sin }^{6}x$

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

${\sin }^{4}x+{\cos }^{4}x-{\sin }^{2}x{\cos }^{2}x$

$1-2{\sin }^{2}x{\cos }^{2}x-{\sin }^{2}x{\cos }^{2}x$

$1-3{\sin }^{2}x{\cos }^{2}x$

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

$3[1-2{\sin }^{2}x{\cos }^2]-2[1-3{\sin }^{2}x{\cos }^{2}x]$

$3-2-6{\sin }^{2}x{\cos }^{2}x+6{\sin }^{2}x{\cos }^{2}x$

$⟹1$