Question

${\displaystyle \frac{x^2\cos \frac{\pi }{4}}{\sin x}}$

Answer 1

${\displaystyle \frac{d}{dx}\left(\frac{x^2\cos \frac{\pi }{4}}{\sin x}\right)=\cos \frac{\pi }{4}\cdot \frac{d}{dx}\left(\frac{x^2}{\sin x}\right)}$

= ${\displaystyle \frac{\frac{1}{\sqrt{2}}[\sin x\cdot \frac{d}{dx}\left(x^2\right)-x^2\cdot \frac{d}{dx}\left(\sin x\right)]}{{\sin }^{2}x}}$   ......[Using quotient rule]

= ${\displaystyle \frac{1}{\sqrt{2}}\left[\frac{\sin x\cdot 2x-x^2\cos x}{{\sin }^{2}x}\right]}$

= ${\displaystyle \frac{1}{\sqrt{2}}[\frac{2x}{\sin x}-\frac{x^2\cos x}{{\sin }^{2}x}]}$

= ${\displaystyle \frac{1}{\sqrt{2}}[2x \text{cosec}x=x^2\cot x \text{cosec}x]}$

= ${\displaystyle \frac{x}{\sqrt{2}} \text{cosec}x[2-x\cot x]}$