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Given cot x and its quadrant: simplify sin A (cos B + sin B) + cos A (cos B - sin B)

The same question, asked 2 times: 23 Jan 2026 S2*, 24 Jan 2026 S1. Every wording below, oldest first.

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  1. Question 1

    23 Jan 2026 S2

    52
    Let π2<θ<π\frac{\pi}{2}< \theta < \pi and cot⁡θ=−122\cot\theta = -\frac{1}{2\sqrt{2}}. Then the value of sin⁡(15θ2)(cos⁡8θ+sin⁡8θ)+cos⁡(15θ2)(cos⁡8θ−sin⁡8θ)\sin\left( \frac{15\theta}{2} \right)(\cos8\theta + \sin8\theta) + \cos\left( \frac{15\theta}{2} \right)(\cos8\theta - \sin8\theta) is equal to :

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  2. Question 2

    24 Jan 2026 S1

    60
    If cot⁡x=512\cot x =\frac{5}{12} for some x∈(π,3π2)x \in\left( \pi,\frac{3\pi}{2} \right), then sin⁡7x(cos⁡13x2+sin⁡13x2)+cos⁡7x(cos⁡13x2−sin⁡13x2)\sin7x\left( \cos\frac{13x}{2}+ \sin\frac{13x}{2} \right)+ \cos7x\left( \cos\frac{13x}{2}- \sin\frac{13x}{2} \right) is equal to

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