CDS 2026 (I) Elementary Mathematics, question 41

Topic: Trigonometry

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Q41

If cos⁡θ1−sin⁡θ+cos⁡θ1+sin⁡θ=4\frac{\cos \theta}{1-\sin \theta}+\frac{\cos \theta}{1+\sin \theta}=4, then which one of the following is a value of (tan⁡2θ+cot⁡2θ)(\tan^{2}\theta+\cot^{2}\theta) ?

  1. (a) 53\frac{5}{3}
  2. (b) 103\frac{10}{3}Correct answer
  3. (c) 44
  4. (d) 55

Solution.

Answer: (b) 103\frac{10}{3}

  1. Combine the two fractions: 2cos⁡θ/(1−sin⁡2θ)=2/cos⁡θ=42\cos\theta/\left( 1 - \sin^{2}\theta \right) = 2/\cos\theta = 4.
  2. Thus cos⁡θ=1/2\cos\theta = 1/2 and tan⁡2θ=(1−1/4)/(1/4)=3\tan^{2}\theta = (1 - 1/4)/(1/4) = 3.
  3. Therefore tan⁡2θ+cot⁡2θ=3+1/3=10/3\tan^{2}\theta + \cot^{2}\theta = 3 + 1/3 = 10/3.