Find the derivative of the function: $\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$

Let $f(x) = \frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$.
Using the quotient rule $\left(\frac{u}{v}\right)^{\prime} = \frac{u^{\prime}v - uv^{\prime}}{v^{2}}$,where $u = x^{2} \cos \left(\frac{\pi}{4}\right)$ and $v = \sin x$:
$f^{\prime}(x) = \cos \left(\frac{\pi}{4}\right) \left[ \frac{\sin x \cdot \frac{d}{dx}(x^{2}) - x^{2} \cdot \frac{d}{dx}(\sin x)}{\sin^{2} x} \right]$
$f^{\prime}(x) = \cos \left(\frac{\pi}{4}\right) \left[ \frac{\sin x(2x) - x^{2}(\cos x)}{\sin^{2} x} \right]$
$f^{\prime}(x) = \frac{x \cos \left(\frac{\pi}{4}\right) (2 \sin x - x \cos x)}{\sin^{2} x}$