The number of solutions of the equation $4 \sin ^{2} x-4 \cos ^{3} x+9-4 \cos x=0 ; x \in[-2 \pi, 2

Question

The number of solutions of the equation $4 \sin ^{2} x-4 \cos ^{3} x+9-4 \cos x=0 ; x \in[-2 \pi, 2 \pi]$ is :\

Professor PC Thomas Classes · Answer

(a): We have,\ $4 \sin ^{2} x-4 \cos ^{3} x+9-4 \cos x=0$\ $\Rightarrow \quad 4\left(1-\cos ^{2} xight)-4 \cos ^{3} x+9-4 \cos x=0$\ $\Rightarrow \quad 4 \cos ^{3} x+4 \cos ^{2} x+4 \cos x=13$\ $\mathrm{I}^{	ext {st }}$ clear that L.H.S. $\leq 12$ and never 13\ $	herefore \quad$ Solution does not exists.