Question

Number of roots of the equation ${\cos }^{7}x+{\sin }^{4}x=1$ in the interval $[0,2\pi ]$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

Solution -

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

$co{s}^{7}x+(1-co{s}^{2}x{)}^2=1$

$co{s}^{7}x+1+co{s}^{4}x-2co{s}^{2}x=1$

$co{s}^{7}x+co{s}^{4}x-2co{s}^{2}x=0$

$co{s}^{5}x+co{s}^{2}x-2=0$

$co{s}^{5}x+co{s}^{2}x=2$

$cosx$ must be equal to 1

because $cosx\le 1$

$cosx=1$

solution = $0,2\pi$

2 solutions

C is correct.