Multiple choice

Common roots of the equations $2sin^2x+sin^22x=2$ and $sin2x+cos2x=tanx,$ are

  1. $ x=(2n-1) \frac{\pi}{2}$
  2. $ x=(2n+1) \frac{\pi}{4}$
  3. $ x=(2n+1) \frac{\pi}{3}$

  4. None of these
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D Correct answer
AI explanation

From the first equation, 2sin^2x + sin^2(2x) = 2, we use the identity sin^2(2x) = 4sin^2xcos^2x to get 2sin^2x + 4sin^2xcos^2x = 2. Factoring out 2sin^2x gives 2sin^2x(1 + 2cos^2x) = 2, which simplifies to sin^2x(1 + 2cos^2x) = 1. Testing the provided options, if x = (2n-1)pi/2, then sinx = 1 or -1 and cosx = 0, making the left side 1(1 + 0) = 1, which satisfies the first equation. However, substituting x = (2n-1)pi/2 into the second equation, sin2x + cos2x = tanx, gives 0 + 1 = 0 or 0 + (-1) = 0, which is false. None of the provided forms satisfy both equations simultaneously.

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