Question

The expression $3(\sin x-cosx{)}^4+6(sinx+cosx{)}^2+4(si{n}^{6}x+co{s}^{6}x)$ is equal to

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

The correct option is D 13

Given expression

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

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

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

$=3[1+{\sin }^{2}2x-2\sin 2x]+6+6\sin 2x+(4-3{\sin }^{2}2x)$

$=13$