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Add fractions with 1 + cos x and 1 - cos x (or 1 + sin) denominators

This type of question, asked 2 times: CDS 1 2017, CDS 1 2020. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 1 2017

    64
    What is the value of sin⁡x11+cos⁡x+11−cos⁡x\sin x \sqrt{\frac{1}{1 + \cos x} + \frac{1}{1 - \cos x}} ?

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  2. Question 2

    CDS 1 2020

    59
    What is cos⁡θ1+sin⁡θ+1cot⁡θ\frac{\cos\theta}{1 + \sin\theta} + \frac{1}{\cot\theta} equal to?

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