Question

The real roots of the equations $co{s}^{7}x+si{n}^{4}x=1$ in the interval $(-\pi ,\pi )$ are ___,_____and_____.

Answer 1

$co{s}^{7}x=1-si{n}^{4}x$
$=(1-si{n}^{2}x)(1+si{n}^{2}x)$
$=co{s}^{2}x(1+si{n}^{2}x)$
$\therefore cosx=0$ or $x=\frac{\pi }{2},-\frac{\pi }{2},$
or $co{s}^{5}x=1+si{n}^{2}x$
$co{s}^{5}x\le 1but1+si{n}^{2}x\ge 1$
$\Rightarrow co{s}^{5}x=1+si{n}^{2}x=1$
$\Rightarrow cosx=1$ and sin x =0.
[both these imply x =0]
Hence, $x=-\frac{\pi }{2},\frac{\pi }{2}and0.$