Question

The value of $3\left[{\sin }^4\left(\frac{3\pi }{2}-x\right)+{\sin }^4\left(3\pi +x\right)\right]-2\left[{\sin }^6\left(\frac{\pi }{2}+x\right)+{\sin }^6\left(5\pi -x\right)\right]$ for all permissibly value of x , is

• A. -1
• B. 0
• C. 1
• D. depends on x

Answer 1

The correct option is C 1

Given

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

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

$⟹{(-\cos x)}^4{(-\sin x)}^4$

$⟹{\sin }^{4}x+{\cos }^{4}x$

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

$⟹1-2{\sin }^{2}x{\cos }^{2}x\cdots \left(1\right)$

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

$⟹{\left(\cos x\right)}^6+{\left(\sin x\right)}^6$

$⟹{\left(\sin x\right)}^6+{\left(\cos x\right)}^6$

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

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

From $\left(1\right)$

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

$1-3{\sin }^{2}x{\cos }^{2}x\cdots \left(2\right)$

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

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

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

$⟹1$