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Exact value of a ratio of 15 or 75 degrees by the compound-angle formula

This type of question, asked 4 times: 17 Jul 2023 S3, 18 Jul 2023 S4, 19 Jul 2023 S2, 25 Jul 2023 S1. All 4 questions below, oldest first.

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  1. Question 1

    17 Jul 2023 S3

    70
    Using cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B, find the value of cos⁡75∘\cos 75^\circ.

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  2. Question 2

    18 Jul 2023 S4

    70
    Using tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}, find the value of tan⁡15∘\tan 15^\circ.

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  3. Question 3

    19 Jul 2023 S2

    61
    Using cot⁡(A−B)=cot⁡Acot⁡B+1cot⁡B−cot⁡A\cot(A - B) = \frac{\cot A\cot B + 1}{\cot B - \cot A}, find the value of cot⁡15∘\cot 15^\circ.

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  4. Question 4

    25 Jul 2023 S1

    52
    Using cosec(α+β)=sec⁡α×sec⁡β×cosec α×cosec βsec⁡α×cosec β+cosec α×sec⁡β\text{cosec}(\alpha + \beta) = \frac{\sec\alpha \times \sec\beta \times \text{cosec}\,\alpha \times \text{cosec}\,\beta}{\sec\alpha \times \text{cosec}\,\beta + \text{cosec}\,\alpha \times \sec\beta}, find the value of cosec 75∘\text{cosec}\,75^\circ.

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