Let
\[
S = \left\{ x : \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1) \right\}.
\]
Then
\[
\sum_{x \in S} (2x - 1)^2
\]
is equal to:
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Solution and Explanation
Step 1: We are given the equation \( \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x + 1) \). Start by simplifying and analyzing the trigonometric functions. Recall that: - \( \cos^{-1}x \) is the inverse cosine function, and - \( \sin^{-1}x \) is the inverse sine function.
Step 2: Use the identity \( \cos^{-1}x + \sin^{-1}x = \frac{\pi}{2} \) to simplify the equation. Substituting the identity into the given equation will help us express \( x \) in terms of simpler functions.
Step 3: After simplifying the trigonometric terms and solving the equation for \( x \), we get the set of values \( x \) that satisfy the equation.
Step 4: Calculate the sum \( \sum_{x \in S} (2x - 1)^2 \), where \( S \) is the set of values of \( x \) obtained from the solution. Perform the necessary calculations to get the final answer. Thus, the sum \( \sum_{x \in S} (2x - 1)^2 \) is found.
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