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NDA 2016 Maths · Mathematics · Trigonometric Equations · medium

For θ∈[0,2π): 1. The equation sin θ=cos θ has exactly two solutions. 2. The equation tan θ=cot θ has exactly the same solutions as sin θ=cos θ.

Single correct option

Consider the following statements for θ∈[0,2π)\theta\in[0,2\pi):
1. The equation sin⁡θ=cos⁡θ\sin\theta=\cos\theta has exactly two solutions.
2. The equation tan⁡θ=cot⁡θ\tan\theta=\cot\theta has exactly the same solutions as sin⁡θ=cos⁡θ\sin\theta=\cos\theta.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (A) 1 only

sin⁡θ=cos⁡θ\sin\theta=\cos\theta means tan⁡θ=1\tan\theta=1, with solutions π4\dfrac\pi4 and 5π4\dfrac{5\pi}4, so statement 1 is true. tan⁡θ=cot⁡θ\tan\theta=\cot\theta gives tan⁡2θ=1\tan^2\theta=1, so tan⁡θ=±1\tan\theta=\pm1 and there are four solutions, including 3π4\dfrac{3\pi}4 and 7π4\dfrac{7\pi}4. Statement 2 is false.
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Original question written in the NDA 2016 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake