If `y=(1/2)^(n-1)cos(ncos^(-1)x),` then prove that `y` satisfies the differential equation `(1-x^2)` `(d^2y)/(dx^2)-x(dy)/(dx)+n^2y=0`

Explanation

To prove that \( y = \left( \frac{1}{2} \right)^{n-1} \cos(n \cos^{-1} x) \) satisfies the differential equation

\[ (1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + n^2 y = 0, \]

we will follow these steps:

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