Question

If `sin (A + B) = cos(A - B) = sqrt(3)/2`, then cot 2A = ______.

Answer 1

If `sin (A + B) = cos(A - B) = sqrt(3)/2`, then cot 2A = 0.

Explanation:

Let S = A + B and D = A – B.

Then `sin S = sqrt(3)/2` ⇒ S = 60° or 120° and `cos D = sqrt(3)/2` ⇒ D = ±30°. 

Now 2A = S + D, so the possible values of 2A are 90°, 30°, 150°, 90°, giving cot 2A = 0, `sqrt(3), -sqrt(3)`, 0. 

Thus cot 2A can be 0 or `±sqrt(3)` in general; under the usual restriction 0° < A + B ≤ 90° and A > B (so S = 60°, D = 30°) we get 2A = 90° and hence cot 2A = 0.