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Is a trig ratio equal to an expression like (a + b)/(2 root ab) possible? AM-GM forces a = b

This type of question, asked 5 times: CDS 2 2018, CDS 2 2020, CDS 1 2021, CDS 2 2023, CDS 1 2025. All 5 questions below, oldest first.

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  1. Question 1

    CDS 2 2018

    66
    Consider the following statements : 1. cos⁡2θ=1−p2+q22pq\cos^2 \theta = 1 - \frac{p^2 + q^2}{2pq}, where p, q are non-zero real numbers, is possible only when p=qp = q. 2. tan⁡2θ=4pq(p+q)2−1\tan^2 \theta = \frac{4pq}{(p + q)^2} - 1, where p, q are non-zero real numbers, is possible only when p=qp = q. Which of the statements given above is/are correct ?

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  2. Question 2

    CDS 2 2020

    49
    Consider the following statements : 1. sin⁡θ=x+1x\sin\theta = x + \dfrac{1}{x} is possible for some real value of x. 2. cos⁡θ=x+1x\cos\theta = x + \dfrac{1}{x} is possible for some real value of x. Which of the above statements is/are correct ?

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  3. Question 3

    CDS 1 2021

    50
    If the equation x2+y2−2xysin⁡2θ=0x^2 + y^2 - 2xy\sin^2 \theta = 0 contains real solution for xx and yy, then

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  4. Question 4

    CDS 2 2023

    6
    For what relation between a and b is the equation sin⁡θ=a+b2ab\sin \theta = \dfrac{a + b}{2\sqrt{ab}} possible ?

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  5. Question 5

    CDS 1 2025

    29
    Which of the following equations is/are possible ? I. sin⁡2θ=(x+y)24xy\sin^2\theta = \frac{(x+y)^2}{4xy}, where x, y are positive unequal real quantities. II. sin⁡θ+cos⁡θ=x+1x\sin\theta + \cos\theta = x + \frac{1}{x}, where x is a positive real quantity. Select the correct answer using the code given below :

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