We'll recall that the tangent function is the
ratio:
tan x = sin x/cos x
tan
(B/2) = sin (B/2)/cos (B/2)
We'll raise to square both
sides:
[tan (B/2)]^2 = [sin (B/2)]^2/[cos
(B/2)]^2
We'll use the half angle
identities:
[sin (B/2)]^2 = (1 - cos
B)/2
[cos (B/2)]^2 = (1 + cos
B)/2
[tan (B/2)]^2 = (1 - cos B)/2/(1 + cos
B)/2
We'll simplify and we'll
get:
[tan (B/2)]^2 = (1 - cos B)/(1 + cos
B)
But, from enunciatin, we know the
followings:
[tan (B/2)]^2 =
(1-sinC)/(1+sinC)
Comparing, we'll
get:
(1 - sinC)/(1 + sinC) = (1 - cos B)/(1 + cos
B)
1 - sin C = 1 - cos B
sin C
= cos B
If sin C = cos B, then the sum of
angles B+C = 90, therefore the measure of the angle A is of 90
degrees.
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