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Given sin x + cos x or sin x cos x, evaluate another expression

This type of question, asked 2 times: 2021 Paper 5, 2021 Paper 16. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    65
    If for x∈(0,π2),log⁡10sin⁡x+log⁡10cos⁡x=−1x\in\left( 0,\frac{\pi}{2} \right),\log_{10}\sin x+\log_{10}\cos x= - 1 and log⁡10(sin⁡x+cos⁡x)=12(log⁡10n−1),n>0\log_{10}(\sin x + \cos x) =\frac{1}{2}\left( \log_{10}n - 1 \right),n > 0, then the value of nn is equal to:

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  2. Question 2

    2021 Paper 16

    47
    If sin⁡θ+cos⁡θ=12\sin\theta+ \cos\theta=\frac{1}{2}, then 16[sin⁡2θ+cos⁡4θ+sin⁡6θ]16\lbrack \sin2\theta+ \cos4\theta+ \sin6\theta\rbrack is equal to

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