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Given one trig ratio and the quadrant, find another ratio or expression

This type of question, asked 2 times: 28 Jun 2022, 8 Apr 2024. All 2 questions below, oldest first.

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  1. Question 1

    28 Jun 2022

    170
    If cot⁡α=1\cot\alpha= 1 and sec⁡β=−53\sec\beta= -\frac{5}{3}, where π<α<3π2\pi<\alpha<\frac{3\pi}{2} and π2<β<π\frac{\pi}{2}<\beta<\pi, then the value of tan⁡(α+β)\tan(\alpha+\beta) and the quadrant in which α+β\alpha+\beta lies, respectively are

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  2. Question 2

    8 Apr 2024

    71
    If sin⁡x=−35\sin x= -\frac{3}{5}, where π<x<3π2\pi<x<\frac{3\pi}{2}, then 80(tan⁡2x−cos⁡x)80\left( \tan^{2}x- \cos x \right) is equal to:

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