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Standard values of compound angles give equations in the angles: solve and evaluate

This type of question, asked 3 times: CDS 2 2018, CDS 1 2022, CDS 2 2024. All 4 questions below, oldest first.

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  1. Question 1

    CDS 2 2018

    64
    Let sin⁡(A+B)=32\sin (A + B) = \frac{\sqrt{3}}{2} and cos⁡B=32\cos B = \frac{\sqrt{3}}{2}, where A, B are acute angles. What is tan⁡(2A−B)\tan (2A - B) equal to ?

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  2. Question 2

    CDS 1 2022

    51
    If AA, BB, CC are acute angles and sin⁡(B+C−A)=cos⁡(C+A−B)=tan⁡(A+B−C)=1\sin(B + C - A) = \cos(C + A - B) = \tan(A + B - C) = 1 then what is (A+B+C)(A + B + C) equal to?

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  3. Question 3

    CDS 1 2022

    47
    If cos⁡(x+y)=0\cos(x + y) = 0 and sin⁡(x−y)=12\sin(x - y) = \frac{1}{2}, where x,y∈[0,π2]x, y \in \left[0, \frac{\pi}{2}\right], then what is the value of cot⁡(2x−y)\cot(2x - y)?

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  4. Question 4

    CDS 2 2024

    14
    If 0≤(α−β)≤(α+β)≤π20 \le (\alpha - \beta) \le (\alpha + \beta) \le \frac{\pi}{2}, tan⁡(α+β)=3\tan(\alpha + \beta) = \sqrt{3} and tan⁡(α−β)=13\tan(\alpha - \beta) = \frac{1}{\sqrt{3}}, then what is tan⁡α⋅cot⁡2β\tan\alpha \cdot \cot 2\beta equal to ?

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