Multiple choice

If $\theta$ is the angle between unit vectors $\hat{A}$ and $\hat{B}$, then $\frac{(1-\hat{A}.\hat{B})}{(1+\hat{A}.\hat{B})}$ is equal to

  1. $tan^{2}\ (\theta/2)$
  2. $sin^{2}\ (\theta/2)$
  3. $cot^{2}\ (\theta/2)$
  4. $cos^{2}\ (\theta/2)$
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A Correct answer
Explanation

The total frequency in a distribution is the sum of frequencies. Given the cumulative frequencies at the boundaries, the total frequency is the sum of the frequencies of all classes. The data provided (less than CF 35 at 60 and greater than CF 25 at 60) implies the total frequency is 35 + 25 = 60.

AI explanation

Since the vectors are unit vectors, their dot product is cos theta. Substituting this into the expression gives (1 - cos theta) divided by (1 + cos theta). Using the half-angle identities, 1 - cos theta is 2 sin^2(theta/2) and 1 + cos theta is 2 cos^2(theta/2). Dividing these terms yields tan^2(theta/2).

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