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Question

The number of values of θ in the interval (π2,π2) such that θnπ5 for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is ___

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Solution

Given, tanθ=cot5θ
tanθ=tan(π25θ)π25θ=nπ+θ6θ=π2nπθ=π12nπ6
Also cos4θ=sin2θ=cos(π22θ)
4θ=2nπ±(π22θ)
Taking positive sign,
6θ=2nπ+π2θ=nπ3+π12
Taking negative sign,
2θ=2nππ2θ=nππ4
Also values of θ suggest that there are only 3 common solutions in the interval of (π2,π2).

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