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a sin X = b sin Y: find a ratio of tangents by componendo and dividendo

This type of question, asked 2 times: 19 Apr 2025 S1, 22 Apr 2025 S2. All 2 questions below, oldest first.

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  1. Question 1

    19 Apr 2025 S1

    110
    If 3sin⁡α=5sin⁡β3\sin\alpha = 5\sin\beta, then tan⁡(α+β2)tan⁡(α−β2)=\dfrac{\tan\left( \frac{\alpha+\beta}{2} \right)}{\tan\left( \frac{\alpha-\beta}{2} \right)}=

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  2. Question 2

    22 Apr 2025 S2

    117
    If sin⁡A=nsin(A+2 B)\sin A = nsin(A + 2\text{ }B), then tan⁡(A+B)=\tan(A + B) =

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