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NDA 2019 Maths · Mathematics · Trigonometric Functions and Identities · medium

If sin θcos θ=1/4, then sin ^4θ+cos ^4θ equals

Single correct option

If sin⁡θcos⁡θ=14\sin\theta\cos\theta=\dfrac14, then sin⁡4θ+cos⁡4θ\sin^4\theta+\cos^4\theta equals
  1. A. 12\dfrac12
  2. B. 1516\dfrac{15}{16}
  3. C. 78\dfrac78
  4. D. 34\dfrac34
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Answer: (C) 78\dfrac78

Write sin⁡4θ+cos⁡4θ=(sin⁡2θ+cos⁡2θ)2−2sin⁡2θcos⁡2θ=1−2(14)2=1−18=78\sin^4\theta+\cos^4\theta=(\sin^2\theta+\cos^2\theta)^2-2\sin^2\theta\cos^2\theta=1-2\left(\dfrac14\right)^2=1-\dfrac18=\dfrac78. The option 1516\dfrac{15}{16} results from using sin⁡2θcos⁡2θ\sin^2\theta\cos^2\theta without the factor 22, and 12\dfrac12 from using sin⁡θcos⁡θ\sin\theta\cos\theta in place of its square.
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Original question written in the NDA 2019 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