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Simplify sin^-1 or cos^-1 of a sin(x + a) type combination on a given interval

This type of question, asked 2 times: 23 Jan 2025, 4 Apr 2025. All 2 questions below, oldest first.

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  1. Question 1

    23 Jan 2025

    64
    If π2≤x≤3π4\frac{\pi}{2}\leq x\leq\frac{3\pi}{4}, then cos⁡−1(1213cos⁡x+513sin⁡x)\cos^{- 1}\left( \frac{12}{13}\cos x+\frac{5}{13}\sin x \right) is equal to

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  2. Question 2

    4 Apr 2025

    63
    Considering the principal values of the inverse trigonometric functions, sin⁡−1(32x+121−x2),−12<x<12\sin^{- 1}\left( \frac{\sqrt{3}}{2}x+\frac{1}{2}\sqrt{1 -x^{2}} \right), -\frac{1}{2}<x<\frac{1}{\sqrt{2}}, is equal to.

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