Question

The number of real solutions of the equation ${\cos }^{7}x+\sin {}^{4}x=1$ the inerval$[-\pi ,\pi ]$is

• A. 2
• B. 3
• C. 5
• D. none of these

Answer 1

The correct option is B 3

Given, $co{s}^{7}x+si{n}^{4}x=1$

$\Rightarrow co{s}^{7}x=1-si{n}^{4}x=(1-si{n}^{2}x)(1+si{n}^{2}x)=co{s}^{2}x(2-co{s}^{2}x)$

$\Rightarrow co{s}^{2}x(co{s}^{5}x+co{s}^{2}x-2)=0$

$cosx=0\Rightarrow x=-\frac{\pi }{2},\frac{\pi }{2}$

$co{s}^{5}x+co{s}^{2}x=2$ we know $-1\le cosx\le 1$

$\Rightarrow$ $cosx=1$ is the only solution $\Rightarrow x=0$
Thus, number of solutions in the given interval is 3.
Hence, option 'B' is correct.