Question

IS the expression :$3{(\sin x-\cos x)}^4+6{(\sin x+\cos x)}^2+4({\sin }^{6}x+{\cos }^{6}x)$ is independent of $x$?.

Answer 1

We have as in $Q.1\left(a\right)$ and $\left(b\right),L.H.S.$
$=3\left[\right(1-2\sin x\cos x{)}^2]+6(1+2\sin x\cos x)+4[({\sin }^{2}x+{\cos }^{2}x{)}^2-3{\sin }^{2}x{\cos }^{2}x]$
$=3(1-2\sin x\cos x{)}^2+6(1+2\sin x\cos x)+4(1-3{\sin }^{2}x+{\cos }^{2}x)$
$=3(1-4\sin x\cos x+4{\sin }^{2}x{\cos }^{2}x{)}^2+6+12\sin x\cos x+4-12{\sin }^{2}x{\cos }^{2}x$
$=13$, which is independent $x$.