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NDA 2025 Maths · Mathematics · Trigonometric Functions and Identities · easy

(for values of θ where the expressions are defined): 1. cos 2θ=1-2sin ^2θ 2. sin 2θ=(2tan θ)/(1+tan ^2θ) Which of the statements is/are correct?

Single correct option

Consider the following statements (for values of θ\theta where the expressions are defined):
1. cos⁡2θ=1−2sin⁡2θ\cos2\theta=1-2\sin^2\theta
2. sin⁡2θ=2tan⁡θ1+tan⁡2θ\sin2\theta=\dfrac{2\tan\theta}{1+\tan^2\theta}
Which of the statements 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: (C) Both 1 and 2

Both are valid. Statement 1 follows from cos⁡2θ=cos⁡2θ−sin⁡2θ\cos2\theta=\cos^2\theta-\sin^2\theta and cos⁡2θ=1−sin⁡2θ\cos^2\theta=1-\sin^2\theta. For statement 2, divide 2sin⁡θcos⁡θ2\sin\theta\cos\theta by sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 and then by cos⁡2θ\cos^2\theta.
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