Question

$3(\sin x-\cos x{)}^4+6(\sin x+\cos x{)}^2+4\left({\sin }^{6}x+{\cos }^{6}x\right)$ is equal to

• A. 11
• B. 12
• C. 13
• D. 14

Answer 1

Answer: (C)

$3(\sin x-\cos x{)}^4+6(\sin x+\cos x{)}^2+4\left({\sin }^{6}x+{\cos }^{6}x\right)$
$=3(1-\sin 2x{)}^2+6(1+\sin 2x)+4\left[{\left({\sin }^{2}x+{\cos }^{2}x\right)}^3-3{\sin }^{2}x{\cos }^{2}x\left({\sin }^{2}x+{\cos }^{2}x\right)\right]$
$=3\left(1-2\sin 2x+{\sin }^{2}2x\right)+(6+6\sin 2x)+\left[1-\frac{3}{4}{\sin }^{2}2x\right]$
$=13+3{\sin }^{2}2x-3{\sin }^{2}2x=13$