CDS 2026 (I) Elementary Mathematics, question 92

Topic: Trigonometry

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Passage · Q91 to Q92

Let 12(tan⁡θ+cot⁡θ)=2512(\tan \theta+\cot \theta)=25, where 45∘<θ<90∘45^{\circ}<\theta<90^{\circ}.

Q92

What is a value of (cosec⁡θ+sec⁡θ)(\operatorname{cosec} \theta+\sec \theta) ?

  1. (a) 44
  2. (b) 3512\frac{35}{12}Correct answer
  3. (c) 2512\frac{25}{12}
  4. (d) 22

Solution.

Answer: (b) 3512\frac{35}{12}

  1. The shared condition gives sin⁡θcos⁡θ=12/25\sin\theta\cos\theta = 12/25.
  2. Also (sin⁡θ+cos⁡θ)2=1+24/25=49/25\left( \sin\theta + \cos\theta \right)^{2} = 1 + 24/25 = 49/25.
  3. Both functions are positive, so sin⁡θ+cos⁡θ=7/5\sin\theta + \cos\theta = 7/5.
  4. Hence cosecθ+sec⁡θ=(sin⁡θ+cos⁡θ)/(sin⁡θcos⁡θ)=(7/5)/(12/25)=35/12cosec\theta + \sec\theta = \left( \sin\theta + \cos\theta \right)/\left( \sin\theta\cos\theta \right) = (7/5)/(12/25) = 35/12.