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NDA 2 2024 Maths, question 27

Topic: Trigonometry

Q27

What is the general solution of cos⁡100x−sin⁡100x=1\cos ^{100}x-\sin ^{100}x=1?

where nn is an integer.

  1. (a) nπn\piCorrect answer
  2. (b) (2n+1)π(2n+1)\pi
  3. (c) 2nπ2n\pi
  4. (d) (2n+1)π/2(2n+1)\pi/2

Solution.

Answer: (a) nπn\pi

  1. Since 0≤cos¹⁰⁰x≤1 and sin¹⁰⁰x≥0, their difference equals 1 only when cos¹⁰⁰x=1 and sin x=0.
  2. Therefore x=nπ for any integer n.
  3. Substitution gives 1−0=1.