Answer
(a) $ x= n\pi, \ \text{or}\ \ x=\frac{3\pi}{4} +n\pi$
(b) $ \bigg\{0,\frac{3\pi}{4} ,\pi,\ \frac{7\pi}{4} \bigg\} $
Work Step by Step
(a) Given
$$( \tan x )( \tan x+1) =0 $$
Then
$$\tan x=0,\ \ \text{or} \ \ \tan x+1=0$$
Case 1 , $$\tan x=0,\ \ \Rightarrow \ \ x=0,\pi ,2\pi,..= n\pi $$
Case 1 , $$\tan x=-1,\ \ \Rightarrow \ \ x=\frac{3\pi}{4} +n\pi $$
Hence the general solutions is
$$ x= n\pi, \ \text{or}\ \ x=\frac{3\pi}{4} +n\pi$$
(b) To find solutions on the interval $ 0\leq x<2\pi $, put values of $n $
$$ \bigg\{0,\frac{3\pi}{4} ,\pi,\ \frac{7\pi}{4} \bigg\} $$
