Question

The no. of real roots of the equation ${\cos }^{7}x+{\sin }^{4}x=1$ in the interval $(-\pi ,\pi )$ are

• A. one
• B. three
• C. two
• D. four

Answer 1

Answer: (B)

three

${\cos }^{7}x+{\sin }^{4}x=1$
${\cos }^{7}x+(1-{\cos }^{2}x{)}^2=1$
${\cos }^{7}x+1+{\cos }^{4}x-2{\cos }^{2}x=1$
${\cos }^{2}x({\cos }^{5}x+{\cos }^{2}x-2)=0$
Thus, $\cos x=0$ or ${\cos }^{5}x+{\cos }^{2}x=2$
$x=\frac{-\pi }{2},\frac{\pi }{2}$, or ${\cos }^{5}x+{\cos }^{2}x=2$, this is possble when both ${\cos }^{5}x={\cos }^{2}x=1$ or $x=0$
Hence, three values are possible for the given equation