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Relations between sums of infinite G.P.s in cos^2 and sin^2 or in a, b, c

This type of question, asked 2 times: 2021 Paper 3, 27 Jun 2022. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    65
    If 0<θ,ϕ<π2,x=∑n=0∞cos⁡2nθ,y=∑n=0∞sin⁡2nϕ0 <\theta,\phi<\frac{\pi}{2},x=\sum_{n= 0}^{\infty} \cos^{2n}\theta,y=\sum_{n= 0}^{\infty} \sin^{2n}\phi and z=∑n=0∞cos⁡2nθ⋅sin⁡2nϕz =\sum_{n = 0}^{\infty} \cos^{2n}\theta\cdot\sin^{2n}\phi then:

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  2. Question 2

    27 Jun 2022

    63
    If x=∑n=0∞an,y=∑n=0∞bn,z=∑n=0∞cnx =\sum_{n = 0}^{\infty} a^{n},y =\sum_{n = 0}^{\infty} b^{n},z =\sum_{n = 0}^{\infty} c^{n}, where a,b,ca,b,c are in A.P. and ∣a∣<1,∣b∣<1,∣c∣<1,abc≠0|a| < 1,|b| < 1,|c| < 1,abc \neq 0, then

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