If $\sin y=x\sin(a+y)$, prove that $\dfrac{dy}{dx}=\dfrac{\sin^{2}(a+y)}{\sin a}$.

CBSE Class 12 · 2012 · DifferentiationLong Answer

sin y = x sin (a + y)

, prove that

\frac{d y}{d x} = \frac{sin ^{2} ( a + y )}{sin a}

Answer: Proved

Write

x = \frac{sin y}{sin ( a + y )}

and differentiate w.r.t.

x

(treat as

1 = \frac{d}{d x}

of the right side via quotient rule, or differentiate

x

w.r.t.

y

\frac{d x}{d y} = \frac{sin ( a + y ) cos y - sin y cos ( a + y )}{sin ^{2} ( a + y )}

The numerator is

sin ((a + y) - y) = sin a

\frac{d x}{d y} = \frac{sin a}{sin ^{2} ( a + y )}

Invert to get

\frac{d y}{d x}

\frac{d y}{d x} = \frac{sin ^{2} ( a + y )}{sin a}

CBSE Class 12 Board · 2012 · Subjective