Question

If $u = \sin (m \cos^{-1} x), v = \cos (m \sin^{-1} x)$, then $\frac{du}{dv} = $

• A. $\sqrt{\frac{ 1- u^2}{ 1- v^2}}$
• B. $\sqrt{\frac{ 1- v^2}{ 1- u^2}}$
• C. $\sqrt{\frac{ 1 + u^2}{ 1 + v^2}}$
• D. none of these.

Answer 1

The Correct Option is A

$u =\sin\left(m \cos^{-1}x\right) $ $\frac{du}{dx} =\cos\left(m \cos^{-1}x\right) . \frac{-m}{\sqrt{1-x^{2}}}$ $ v = \cos\left(m\sin^{-1}x\right) $ $\frac{dv}{dx} =-\sin \left(m\sin^{-1} x\right). \frac{m}{\sqrt{1-x^{2}}}$ $ \therefore \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{\cos\left(m \cos^{-1}x\right)}{\sin\left(m \sin^{-1}x\right)} $ $= \frac{\sqrt{1-\sin^{2}\left(m \cos^{-1}x\right)}}{\sqrt{1-\cos^{2}\left(m\sin^{-1}x\right)}} = \sqrt{\frac{1-u^{2}}{1-v^{2}}} $