Solve for x the equation tan(x+pi/3)=tan(pi/2-x), if 0

The equation could be solved in 2 ways, at
least.

(x + pi/3) = arctan[tan(pi/2-x)] +
kpi

But tan (arctan x) = x

(x
+ pi/3) = (pi/2-x) + kpi

We'll add x both
sides:

2x = pi/2 - pi/3 +
kpi

2x = pi/6 + kpi

x = pi/12
+ kpi

Since x has to belong to the opened
interval (0,pi), then x = pi/12.