Question

Tan(x/2)=sin(x/2) what is x

Answer 1

tan(x/2) = sin(x/2)

sin(x/2) / cos(x/2) = sin(x/2)

sin(x/2) / cos(x/2) - sin(x/2) = 0

sin(x/2) (1/cos(x/2) - 1) = 0

Then either

sin(x/2) = 0   or   1/cos(x/2) - 1 = 0

• If sin(x/2) = 0, we have

x/2 = arcsin(0) + 2nπ   or   x/2 = π - arcsin(0) + 2nπ

(where n is any integer)

x/2 = 2nπ   or   x/2 = π + 2nπ

x = 4nπ   or   x = 2π + 4nπ

• If 1/cos(x/2) - 1 = 0, we have

1/cos(x/2) = 1

cos(x/2) = 1

x/2 = arccos(1) + 2nπ

x/2 = 2nπ

x = 4nπ

but we already have this family of solutions accounted for.