# Let $x = \cos\theta$, $y = \sin\theta$ then  $\frac{d^{2}y}{dx^{2}} =$ ? Option 1) $cosec ^{3}\theta$ Option 2) $- cosec ^{3}\theta$ Option 3) $\sin ^{3}\theta$ Option 4) $-\sin ^{3}\theta$

H Himanshu

As we have learnt,

Second order derivative for parametric function -

When we find

$\frac{dy}{dx}=F(t)\:then\:\frac{d^{2}y}{dx^{2}}=\frac{\frac{d}{dt}\:F(t)}{\frac{dx}{dt}}$

-

$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}= \frac{\frac{d}{d\theta}\sin\theta}{\frac{d}{d\theta}\cos\theta} = \frac{\cos\theta}{-\sin\theta} = -\cot \theta$

$\frac{d^2 y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx} \right ) = \frac{\frac{d}{d\theta}(-\cot\theta)}{\frac{dx}{d\theta}} = \frac{cosec^2 \theta}{-\sin\theta} = -cosec^3\theta$

Option 1)

$cosec ^{3}\theta$

Option 2)

$- cosec ^{3}\theta$

Option 3)

$\sin ^{3}\theta$

Option 4)

$-\sin ^{3}\theta$

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