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If $y=\cos ^{-1} x$, Find $\frac{d^2 y}{d x^2}$ in terms of $y$ alone.
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Verified Answer
$\frac{d y}{d x}=-\left(1-x^2\right)^{-\frac{1}{2}}$
and $\frac{d^2 y}{d x^2}=\frac{-x}{\left(1-x^2\right)^{3 / 2}}=\frac{-\cos y}{\left(\sin ^2 y\right)^{3 / 2}}$
$=-\cot y \operatorname{cosec}^2 y$
and $\frac{d^2 y}{d x^2}=\frac{-x}{\left(1-x^2\right)^{3 / 2}}=\frac{-\cos y}{\left(\sin ^2 y\right)^{3 / 2}}$
$=-\cot y \operatorname{cosec}^2 y$
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