What are the zeros of the function f(x)=sinx+cosx found in interval (0,pi)?

First, we'll put sinx+cosx =
0.

We'll divide by cos x:

sin
x/cos x + 1 = 0

We know that sinx/cosx=tan
x

tanx + 1 = 0

tan x =
-1

x = arctan(-1)

x = -arctan
1

Interval(0,pi) covers boths quadrants, 1st and the
2nd.

The tangent function has positive values in the 1st
quadrant and negative values in the 2nd quadrant.

x = pi -
pi/4 = 3pi/4

The solution of the equation is
x = 3pi/4.