Multiple choice

If $\theta$ is the angle between unit vectors $\bar {A}$ and $\bar {B}$, then $\dfrac{(1-\bar {A}.\bar {B})}{(1+\bar {A}.\bar {B})}$ 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

For unit vectors A and B, A.B = cos(theta). The expression becomes (1 - cos(theta)) / (1 + cos(theta)). Using trigonometric identities, 1 - cos(theta) = 2*sin^2(theta/2) and 1 + cos(theta) = 2*cos^2(theta/2). The ratio is (2*sin^2(theta/2)) / (2*cos^2(theta/2)) = tan^2(theta/2).

AI explanation

Since the vectors are unit vectors, their dot product equals cos theta. The expression becomes (1 - cos theta) divided by (1 + cos theta). Applying the half-angle formulas, the numerator is 2 sin^2(theta/2) and the denominator is 2 cos^2(theta/2). Simplifying this fraction results in tan^2(theta/2).

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