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One trig ratio (or the sides of a right triangle) given: evaluate an expression in the others

This type of question, asked 8 times: CDS 2 2017, CDS 2 2018, CDS 2 2019, CDS 1 2020, CDS 2 2021, CDS 2 2022, CDS 1 2023, CDS 2 2025. All 10 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2017

    46
    If a triangle has sides 5, 13 and 12 units and θ\theta is the acute angle of the triangle, then what is the value of (sin⁡θ+cos⁡θ)(\sin \theta + \cos \theta) ?

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  2. Question 2

    CDS 2 2017

    48
    If sin⁡θ=m2−n2m2+n2\sin \theta = \frac{m^2 - n^2}{m^2 + n^2} and 0<θ<π20 < \theta < \frac{\pi}{2}, then what is the value of cos⁡θ\cos \theta ?

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  3. Question 3

    CDS 2 2018

    61
    If cos⁡θ=15\cos \theta = \frac{1}{\sqrt{5}}, where 0<θ<π20 < \theta < \frac{\pi}{2}, then 2tan⁡θ1−tan⁡2θ\frac{2 \tan \theta}{1 - \tan^2 \theta} is equal to

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  4. Question 4

    CDS 2 2019

    51
    If tan⁡x=1\tan x = 1, 0<x<90∘0 < x < 90^\circ, then what is the value of 2sin⁡xcos⁡x2 \sin x \cos x ?

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  5. Question 5

    CDS 1 2020

    63
    If θ\theta lies in the first quadrant and cot⁡θ=6316\cot\theta = \frac{63}{16}, then what is the value of (sin⁡θ+cos⁡θ)(\sin\theta + \cos\theta) ?

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  6. Question 6

    CDS 1 2020

    65
    A rectangle is 48 cm long and 14 cm wide. If the diagonal makes an angle θ\theta with the longer side, then what is (sec⁡θ+cosec⁡θ)(\sec\theta + \operatorname{cosec}\theta) equal to ?

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  7. Question 7

    CDS 2 2021

    57
    If for some θ\theta lying between 0∘0^\circ and 90∘90^\circ, tan⁡θ=1\tan\theta=1, then what is the value of sin⁡2θ−2sin⁡θcos⁡θ\sin^2\theta-2\sin\theta\cos\theta ?

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  8. Question 8

    CDS 2 2022

    51
    If tan⁡θ=q2−p2p\tan\theta = \dfrac{\sqrt{q^2 - p^2}}{p}; 0<θ<90∘0 < \theta < 90^\circ, then what is sec⁡θ+cos⁡θ+2\sec\theta + \cos\theta + 2 equal to ?

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  9. Question 9

    CDS 1 2023

    47
    If sin⁡θ=1213\sin\theta = \frac{12}{13}, then what is the value of (tan⁡θ+sec⁡θ)2(cosec⁡ θ−cot⁡θ)−2(\tan\theta + \sec\theta)^2 (\operatorname{cosec}\,\theta - \cot\theta)^{-2}, 0<θ<π20 < \theta < \frac{\pi}{2} ?

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  10. Question 10

    CDS 2 2025

    35
    If cot⁡θ=7\cot\theta = \sqrt{7}, then what is cosec2θ−sec⁡2θcosec2θ+sec⁡2θ\dfrac{\mathrm{cosec}^2\theta - \sec^2\theta}{\mathrm{cosec}^2\theta + \sec^2\theta} equal to?

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