How many solutions has the equation tg x - 1 = 0 , over the interval ( 0 , 2pi ) ?

This is a trigonometric elementary equation.

We'll move the free term to the right side:

tg x = 1

The function tangent is positive in the first and the third quadrants.

Now, let's find out for what values of angles, the tangent has the value 1.

x = arctg 1 + k*pi

x = pi/4 + k*pi

If k = 0, x = pi/4 and it is an angle located in the first quadrant.

If k = 0, x = pi/4 + pi and it is an angle located in the third quadrant.

So, over the interval (0, 2pi), the equation will have only 2 solutions, namely: {pi/4}U{pi + pi/4}.