Question

Find the real roots of the equation ${\cos }^{7}x+{\sin }^{4}x=1$ in the interval $(-\pi ,\pi )$ .

Answer 1

Ans.$-\pi /2,0,\pi /2$
$co{s}^{7}x=1-{\sin }^{4}x=(1-{\sin }^{2}x)(1+{\sin }^{2}x)$
$={\cos }^{2}x(1+{\sin }^{2}x)$
$\therefore \cos x=0$ or $x=\pi /2,-\pi /2$
or ${\cos }^{5}x=1+{\sin }^{2}x$ or ${\cos }^{5}x-{\sin }^{2}x=1$
Now maximum value of each $\cos x$ or $\sin x$ is $1$. Hence the above equation will hold when $\cos x=1$ and $\sin x=0$. Both these imply $x=0$.
Hence $x=\frac{\pi }{2},\frac{\pi }{2},0$