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Exponential equation in sin^2 x and cos^2 x (or tan^2 x): put t = a^(sin^2 x)

This type of question, asked 4 times: 2021 Paper 5, 2021 Paper 23, 26 Jul 2022, 10 Apr 2023. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    79
    The number of roots of the equation,
    (81)sin⁡2x+(81)cos⁡2x=30(81)^{\sin^{2}x}+ (81)^{\cos^{2}x}= 30
    in the interval [0,π]\lbrack 0,\pi\rbrack is equal to:

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  2. Question 2

    2021 Paper 23

    77
    The number of solutions of the equation 32tan⁡2x+32sec⁡2x=81,0≤x≤π432^{\tan^{2}x}+32^{\sec^{2}x}= 81,0 \leq x\leq\frac{\pi}{4} is:

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  3. Question 3

    26 Jul 2022

    78
    Let S={θ∈[0,2π]:82sin⁡2θ+82cos⁡2θ=16}S =\left\{ \theta \in \lbrack 0,2\pi\rbrack:8^{2\sin^{2}\theta}+8^{2\cos^{2}\theta}= 16 \right\}. Then n(S)+∑θ∈S(sec(π4+2θ)csc⁡(π4+2θ))n(S) +\sum_{\theta \in S} \left( sec\left( \frac{\pi}{4}+ 2\theta \right)\csc\left( \frac{\pi}{4}+ 2\theta \right) \right) is equal to:

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  4. Question 4

    10 Apr 2023

    165
    Let S={x∈(−π2,π2):91−tan⁡2x+9tan⁡2x=10}S =\left\{ x \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right):9^{1 -\tan^{2}x}+9^{\tan^{2}x}= 10 \right\} and β=∑x∈Stan⁡2(x3)\beta =\sum_{x \in S} \tan^{2}\left( \frac{x}{3} \right), then 16(β−14)2\frac{1}{6}(\beta - 14)^{2} is equal to

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