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NDA 1 2024 Maths, question 47

Topic: Trigonometry

Q47

If f(θ)=11+tan⁡θf(\theta)=\frac{1}{1+\tan \theta} and α+β=5π4\alpha+\beta=\frac{5\pi}{4}, then what is the value of f(α)f(β)f(\alpha)f(\beta) ?

  1. (a) −12-\frac{1}{2}
  2. (b) 12\frac{1}{2}Correct answer
  3. (c) 11
  4. (d) 22

Solution.

Answer: (b) 12\frac{1}{2}

  1. Let u = tan α and v = tan β.
  2. Since tan(α + β) = 1, u + v = 1 − uv.
  3. Hence (1 + u)(1 + v) = 2, so f(α)f(β) = 1/2 wherever the given functions are defined.