Question

If $y={\cos }^{-1}\left(\frac{2\cos x+3\sin x}{\sqrt{\left(13\right)}}\right)$ then $\frac{dy}{dx}=$

Answer 1

Put $2=r\cos a,3=r\sin \alpha$
$\therefore r=\sqrt{13},\tan \alpha =3/2.$
$\therefore y={\cos }^{-1}(\cos x\cos \alpha +\sin x\sin \alpha )$
$={\cos }^{-1}\cos (x-\alpha )=x-\alpha$
or $y=x-{\tan }^{-1}\frac{3}{2}\therefore \frac{dy}{dx}=1.$