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If cot^(-1)(sqrt(cos(alpha))) - tan^(-1)(sqrt(cos(alpha))) = x, then which of the following relations is correct?

1. sin(x) = cot²(alpha/2)
2. cos(x) = 2*sqrt(cos(alpha)) / (1 - cos(alpha))
3. sin(x) = tan²(alpha/2)
4. cos(x) = 2*cos(alpha) / (1 + cos(alpha))

Correct answer: sin(x) = tan²(alpha/2)

Solution

Substituting cot^(-1)(t) = pi/2 - tan^(-1)(t) gives x = pi/2 - 2*tan^(-1)(sqrt(cos(alpha))). So sin(x) = cos(2*tan^(-1)(sqrt(cos(alpha)))). Using cos(2*theta) = (1-tan²(theta))/(1+tan²(theta)) with tan(theta) = sqrt(cos(alpha)): sin(x) = (1 - cos(alpha))/(1 + cos(alpha)) = tan²(alpha/2).

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