Question

Differentiate w.r.t $x$.
${\sin }^{-1}\left(\frac{a\cos x+b\sin x}{\sqrt{a^2+b^2}}\right)$

Answer 1

Let $y={\sin }^{-1}\left(\frac{a\cos x+b\sin x}{\sqrt{a^2+b^2}}\right)$
Let $\left(\frac{a}{\sqrt{a^2+b^2}}\right)=\sin \alpha$
Then, $\frac{b}{\sqrt{a^2+b^2}}=\cos \alpha$
$y={\sin }^{-1}(\sin \alpha \cos x+\cos \alpha \sin x)$
$={\sin }^{-1}\left\{\sin \right(x+\alpha \left)\right\}$
$y=x+\alpha$
Differentiating w.r.t $x$, we get
$\frac{dy}{dx}=1.[\alpha \to$ constant $]$