2 / cos(A/2)
We use identity:
cos B + cos C = 2 cos((B + C)/2) cos((B - C)/2)
Since A + B + C = 180° ⇒ B + C = 180° − A
So,
cos((B + C)/2) = cos((180° − A)/2) = cos(90° − A/2) = sin(A/2)
Thus,
cos B + cos C = 2 sin(A/2) cos((B − C)/2)
Given sin A = 2 sin(A/2) cos((B − C)/2)
Also,
sin A = 2 sin(A/2) cos(A/2)
Comparing both equations:
cos((B − C)/2) = cos(A/2)
⇒ B = C
Now B = C ⇒ B + C = 180° − A
2B = 180° − A
B = 90° − A/2
Now find:
tan(B/2) + cot(B/2)
Let x = B/2
Then expression becomes:
tan x + cot x
Identity:
tan x + cot x = 1/(sin x cos x) = 2/sin(2x)
So,
= 2/sin B
Now substitute B = 90° − A/2:
sin B = sin(90° − A/2) = cos(A/2)
Final answer:
tan(B/2) + cot(B/2) = 2 / cos(A/2)
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