The number of solutions of the equation x e^sin x - cos x = 0 in the…

Let $f (x) = x\, e^{\sin x} - cos x$. Then,

$f'(x)=e^{\sin x}+x\, e^{\sin x} \cos x + \sin x > 0$ for all $x ∈ (0, π/2)$

⇒ f(x) is increasing on (0, π/2)

Also, $f (0) = -1$ and $f(\frac{π}{2})=\frac{π}{2}e$

So, there is a point between 0 and π/2, where $f(x) = 0$.

Hence, $f(x) = 0$ has exactly one solution in (0, π/2).