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Right triangle: an expression in two acute angles folds to sin, cos or cot of their sum or difference

This type of question, asked 3 times: 21 Jul 2023 S3, 26 Jul 2023 S2, 11 Sep 2024 S1. All 3 questions below, oldest first.

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  1. Question 1

    21 Jul 2023 S3

    55
    In △XYZ\triangle XYZ, right angled at Y, if sin⁡X=12\sin X = \frac{1}{2}, find the value of cos⁡Xcos⁡Z+sin⁡Xsin⁡Z\cos X \cos Z + \sin X \sin Z.

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  2. Question 2

    26 Jul 2023 S2

    59
    ΔABC\Delta ABC is a right triangle. If ∠B=90∘\angle B = 90^\circ and tan⁡A=13\tan A = \frac{1}{\sqrt{3}}, then the value of sin⁡Acos⁡C+cos⁡Asin⁡C\sin A \cos C + \cos A \sin C is:

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  3. Question 3

    11 Sep 2024 S1

    65
    In a △ABC\triangle ABC right-angled at C, if tan⁡A=3\tan A = \sqrt{3}, then the value of sin⁡Acos⁡Bcot⁡(A+B)\sin A \cos B \cot(A + B) is:

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