Mathematics - Trigonometric Equation Question with Solution | TestHub

$16\left({\sin }^{5}x+{\cos }^{5}x\right)-11(\sin x+\cos x)=0$

$\begin{array}{rl}& \Rightarrow (\sin +\cos x)\left\{16\left({\sin }^{4}x-{\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x-\sin x{\cos }^{3}x+{\cos }^{4}x\right)-11\right\}=0\\ & \Rightarrow (\sin x+\cos x)\left\{16\left(1-{\sin }^{2}x{\cos }^{2}x-\sin x\cos x\right)-11\right\}=\\ & \Rightarrow (\sin x+\cos x)(4\sin x\cos x-1)(4\sin x\cos x+5)=0\end{array}$

As $4\sin x\cos x+5\ne 0$, WE HAVE

$\sin x+\cos x=0,4\sin x\cos x-1=0$

THE REQUIRED VALUES ARE $\pi /12,5\pi /12,9\pi /12,13\pi /12,17\pi /12,21\pi /12$,

THEY ARE 6 SOLUTIONS ON $[0,2\pi ]$