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Sum of cosines equals its maximum, so each cosine is 1

This type of question, asked 2 times: CDS 1 2017, CDS 1 2021. All 2 questions below, oldest first.

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  1. Question 1

    CDS 1 2017

    62
    If cos⁡θ1+cos⁡θ2+cos⁡θ3=3\cos \theta_1 + \cos \theta_2 + \cos \theta_3 = 3, then what is sin⁡θ1+sin⁡θ2+sin⁡θ3\sin \theta_1 + \sin \theta_2 + \sin \theta_3 equal to ?

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  2. Question 2

    CDS 1 2021

    55
    Let cos⁡α+cos⁡β=2\cos\alpha + \cos\beta = 2 and sin⁡α+sin⁡β=0\sin\alpha + \sin\beta = 0, where 0≤α≤90∘0 \le \alpha \le 90^\circ, 0≤β≤90∘0 \le \beta \le 90^\circ. What is the value of cos⁡2α−cos⁡2β\cos 2\alpha - \cos 2\beta?

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