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Simplify expressions in sin^4 + cos^4 and sin^6 + cos^6

This type of question, asked 4 times: CDS 2 2019, CDS 1 2020, CDS 2 2022, CDS 2 2023. All 4 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2019

    59
    What is the value of sin⁡6θ+cos⁡6θ+3sin⁡2θcos⁡2θ−1\sin^6 \theta + \cos^6 \theta + 3 \sin^2 \theta \cos^2 \theta - 1 ?

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  2. Question 2

    CDS 1 2020

    64
    What is the value of 1−2sin⁡2θcos⁡2θsin⁡4θ+cos⁡4θ+4\frac{1 - 2\sin^2\theta\cos^2\theta}{\sin^4\theta + \cos^4\theta} + 4 equal to ?

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  3. Question 3

    CDS 2 2022

    58
    What is 2sin⁡6θ+2cos⁡6θ−3sin⁡4θ−3cos⁡4θ2\sin^6\theta + 2\cos^6\theta - 3\sin^4\theta - 3\cos^4\theta equal to ?

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  4. Question 4

    CDS 2 2023

    25
    What is 3(sin⁡x−cos⁡x)4+6(sin⁡x+cos⁡x)2+4(sin⁡x)6+4(cos⁡x)63(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin x)^6 + 4(\cos x)^6 equal to ?

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