Question

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

• A. $0,\pi /2,-\pi /2$
• B. 0
• C. $-\pi /2,\pi /2$
• D. $\pi /2$

Answer 1

The correct option is A $0,\pi /2,-\pi /2$

${\cos }^{7}x+{\sin }^{4}x=1\Rightarrow {\cos }^{7}x+{(1-{\cos }^{2}x)}^2=1\Rightarrow {\cos }^{7}x+1+{\cos }^{4}x-2{\cos }^{2}x=1\Rightarrow {\cos }^{7}x+{\cos }^{4}x-2{\cos }^{2}x=0\Rightarrow {\cos }^{2}x({\cos }^{5}x+{\cos }^{2}x-2)=0$
$\Rightarrow {\cos }^{2}x=0$ or ${\cos }^{5}x+{\cos }^{2}x-2=0$
$\Rightarrow x=\pm \frac{\pi }{2}$ or $x=0$