Find d²y/dx², if y=2 sin⁻¹⁡(cos⁡x)

The correct option is 0.

Let us find the second derivative of the given function step-by-step.
The given function is:
$$y=2{\sin }^{-1}(\cos x)$$

We can simplify this function by using trigonometric identities. Recall the co-function identity that relates sine and cosine:
$$\cos x=\sin (\frac{\pi }{2}-x)$$

Substituting this identity into our expression for $y$, we get:
$$y=2{\sin }^{-1}\left[\sin \right(\frac{\pi }{2}-x\left)\right]$$

Since the inverse sine function cancels the sine function within its principal domain, the equation simplifies to:
$$y=2(\frac{\pi }{2}-x)$$
Multiplying the terms inside the parentheses by 2, we obtain:
$$y=\pi -2x$$

Now, we differentiate $y$ with respect to $x$ to find the first derivative:
$$\frac{dy}{dx}=\frac{d}{dx}(\pi -2x)$$
Since $\pi$ is a constant, its derivative is 0, and the derivative of $-2x$ is $-2$:
$$\frac{dy}{dx}=-2$$

To find the second derivative, we differentiate the first derivative with respect to $x$ once more:
$$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(-2)$$
Since $-2$ is a constant, its derivative is 0:
$$\frac{d^{2}y}{d{x}^2}=0$$