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NDA 2016 Maths · Mathematics · Trigonometric Functions and Identities · medium

For all real θ for which the expressions are defined, consider the following statements: 1. sec ^2θ+cosec^2θ=sec ^2thetacosec^2θ. 2. sin ^4θ-cos…

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For all real θ\theta for which the expressions are defined, consider the following statements:
1. sec⁡2θ+cosec⁡2θ=sec⁡2θcosec⁡2θ\sec^2\theta+\operatorname{cosec}^2\theta=\sec^2\theta\operatorname{cosec}^2\theta.
2. sin⁡4θ−cos⁡4θ=sin⁡2θ−cos⁡2θ\sin^4\theta-\cos^4\theta=\sin^2\theta-\cos^2\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
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Answer: (C) Both 1 and 2

Statement 1: sec⁡2θ+cosec⁡2θ=sin⁡2θ+cos⁡2θsin⁡2θcos⁡2θ=sec⁡2θcosec⁡2θ\sec^2\theta+\operatorname{cosec}^2\theta=\dfrac{\sin^2\theta+\cos^2\theta}{\sin^2\theta\cos^2\theta}=\sec^2\theta\operatorname{cosec}^2\theta, true. Statement 2: sin⁡4θ−cos⁡4θ=(sin⁡2θ−cos⁡2θ)(sin⁡2θ+cos⁡2θ)=sin⁡2θ−cos⁡2θ\sin^4\theta-\cos^4\theta=(\sin^2\theta-\cos^2\theta)(\sin^2\theta+\cos^2\theta)=\sin^2\theta-\cos^2\theta, true. Both hold.
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