If `x in (0, (pi)/(2))`, then show that `cos^(-1) ((7)/(2) (1 + cos 2 x) + sqrt((sin^(2) x - 48 cos^(2) x)) sin x) = x - cos^(-1) (7 cos x)`

Explanation

To prove the equation

\[ \cos^{-1} \left( \frac{7}{2} (1 + \cos 2x) + \sqrt{\sin^2 x - 48 \cos^2 x} \sin x \right) = x - \cos^{-1} (7 \cos x) \]

for \( x \in (0, \frac{\pi}{2}) \), we will follow these steps:

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