Question

Find the number of real solutions of the equation ${\left(\cos x\right)}^5+{\left(\sin x\right)}^3=1$ in the interval $[0,2\pi ]$

Answer 1

Given, ${\cos }^{5}x+{\sin }^{3}x=1\Rightarrow {\cos }^{3}x(1-{\sin }^{2}x)+{\sin }^{3}x=1\Rightarrow {\cos }^{3}x-{\cos }^{3}x{\sin }^{2}x+{\sin }^{3}x=1\Rightarrow {\sin }^{2}x{\cos }^{3}x=0$
$\Rightarrow {\sin }^{2}x=0$ or ${\cos }^{3}x=0$
$\Rightarrow \sin x=0$ or $\cos x=0$
$\Rightarrow x=n\pi$ or $x=\frac{(2n+1)\pi }{2}$