Multiple choice

If $\cot 2\theta=\sqrt 3$ and $\theta$ is acute angle, then $\sin^2(45+\theta)-\cos^2(75-\theta)$ is

  1. 1
  2. $\dfrac {\sqrt 3-1}{4}$
  3. $\dfrac {1}{2}$
  4. $\dfrac {\sqrt 3-2}{4}$
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C Correct answer
Explanation

cot 2theta = sqrt(3) implies 2theta = 30 degrees, so theta = 15 degrees. Expression: sin^2(45+15) - cos^2(75-15) = sin^2(60) - cos^2(60) = (sqrt(3)/2)^2 - (1/2)^2 = 3/4 - 1/4 = 2/4 = 1/2.

AI explanation

Given cot 2 theta equals the square root of 3, 2 theta equals 30 degrees, so theta is 15 degrees. The expression becomes sin squared of 60 degrees minus cos squared of 60 degrees. Using the identity sin squared x minus cos squared x equals negative cos of 2x, this simplifies to negative cos of 120 degrees. Since the cosine of 120 degrees is negative one half, the result is one half.

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