CDS 2026 (I) Elementary Mathematics, question 43

Topic: Trigonometry

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Q43

If 3sin⁡θ+4cos⁡θ=53\sin \theta+4\cos \theta=5, then what is a value of 4tan⁡θ+3cot⁡θ4\tan \theta+3\cot \theta ?

  1. (a) 77Correct answer
  2. (b) 88
  3. (c) 99
  4. (d) 1010

Solution.

Answer: (a) 77

  1. Use (3sin⁡θ+4cos⁡θ)2+(4sin⁡θ−3cos⁡θ)2=25\left( 3\sin\theta + 4\cos\theta \right)^{2} + \left( 4\sin\theta - 3\cos\theta \right)^{2} = 25.
  2. The given first term already equals 25, so 4sin⁡θ−3cos⁡θ=04\sin\theta - 3\cos\theta = 0.
  3. Hence tan⁡θ=3/4\tan\theta = 3/4 and cot⁡θ=4/3\cot\theta = 4/3, giving 4tan⁡θ+3cot⁡θ=3+4=74\tan\theta + 3\cot\theta = 3 + 4 = 7.