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sin + sin^2 = 1 (or cos, tan, cot form): evaluate a sum of powers of the other ratio

This type of question, asked 5 times: 19 Jul 2023 S3, 26 Jul 2023 S1, 27 Jul 2023 S3, 12 Sep 2024 S2, 23 Sep 2024 S2. All 5 questions below, oldest first.

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  1. Question 1

    19 Jul 2023 S3

    59
    For any acute angle θ\theta, sin⁡θ+sin⁡2θ=1\sin \theta + \sin^2 \theta = 1, then the value of cos⁡2θ+cos⁡4θ=\cos^2 \theta + \cos^4 \theta = ________.

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  2. Question 2

    26 Jul 2023 S1

    62
    If cos⁡A+cos⁡2A=1\cos A + \cos^2 A = 1, then the value of sin⁡4A+sin⁡6A\sin^4 A + \sin^6 A is:

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  3. Question 3

    27 Jul 2023 S3

    65
    If tan⁡4θ+tan⁡2θ=1\tan^4\theta + \tan^2\theta = 1, what is the value of 11(cos⁡4θ+cos⁡2θ)11(\cos^4\theta + \cos^2\theta)?

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  4. Question 4

    12 Sep 2024 S2

    72
    If cot⁡4ϕ−cot⁡2ϕ=1\cot^4\phi - \cot^2\phi = 1, then the value of cos⁡4ϕ+cos⁡2ϕ\cos^4\phi + \cos^2\phi is:

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  5. Question 5

    23 Sep 2024 S2

    68
    If cos⁡A+cos⁡2A=1\cos A + \cos^2 A = 1 then sin⁡2A+sin⁡4A\sin^2 A + \sin^4 A is equal to:

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