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2/(tgx+1)<2-tgx inequation

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    2                  
---------- < 2 - tan(x)
tan(x) + 1

$$\frac{2}{\tan{\left(x \right)} + 1} < 2 - \tan{\left(x \right)}$$

2/(tan(x) + 1) < 2 - tan(x)

Solving inequality on a graph

Rapid solution 2 [src]

    pi     pi  3*pi 
(0, --) U (--, ----)
    4      2    4

$$x\ in\ \left(0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{2}, \frac{3 \pi}{4}\right)$$

x in Union(Interval.open(0, pi/4), Interval.open(pi/2, 3*pi/4))

Rapid solution [src]

  /   /           pi\     /pi          3*pi\\
Or|And|0 < x, x < --|, And|-- < x, x < ----||
  \   \           4 /     \2            4  //

$$\left(0 < x \wedge x < \frac{\pi}{4}\right) \vee \left(\frac{\pi}{2} < x \wedge x < \frac{3 \pi}{4}\right)$$

((0 < x)∧(x < pi/4))∨((pi/2 < x)∧(x < 3*pi/4))