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Telescoping sums of trig terms

This type of question, asked 2 times: 28 Jan 2025, 4 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    28 Jan 2025

    141
    If ∑r=113{1sin⁡(π4+(r−1)π6)sin⁡(π4+rπ6)}=a3+b\sum_{r = 1}^{13} \left\{ \frac{1}{\sin\left( \frac{\pi}{4}+ (r - 1)\frac{\pi}{6} \right)\sin\left( \frac{\pi}{4}+\frac{r\pi}{6} \right)} \right\}= a\sqrt{3}+ b, a,b∈Za,b \in Z, then a2+b2a^{2}+b^{2} is equal to :

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  2. Question 2

    4 Apr 2026 S1

    73
    If A=sin⁡3∘cos⁡9∘+sin⁡9∘cos⁡27∘+sin⁡27∘cos⁡81∘A = \frac{\sin 3^{\circ}}{\cos 9^{\circ}} + \frac{\sin 9^{\circ}}{\cos 27^{\circ}} + \frac{\sin 27^{\circ}}{\cos 81^{\circ}} and B=tan⁡81∘−tan⁡3∘B = \tan81^{\circ} - \tan3^{\circ}, then BA\frac{B}{A} is equal to ____\_\_\_\_ .