Given the ranges of $\displaystyle \displaystyle \alpha \text{ and } \beta$, clearly both $\displaystyle \displaystyle \tan \alpha , ~ \tan \beta > 0$, and so, their sum will be positive also. That is, $\displaystyle \displaystyle 0 < \tan \alpha + \tan \beta$.
Now it remains only to show that $\displaystyle \displaystyle \tan \alpha + \tan \beta < 1$. Assume, to the contrary, that $\displaystyle \displaystyle \tan \alpha + \tan \beta \ge 1$.
Since $\displaystyle \displaystyle \alpha = \frac {\pi}4 - \beta$, this is the same as saying
$\displaystyle \displaystyle \tan \left( \frac {\pi}4 - \beta \right) + \tan \beta \ge 1$
Now, find the contradiction
Introduce tan on both sides of the given equation:
$\displaystyle \tan(\alpha + \beta) = \tan \dfrac{\pi}{4}$.
This becomes:
$\displaystyle \dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}= 1$
$\displaystyle \tan\alpha + \tan\beta = 1 - \tan\alpha\tan\beta $
Now, since $\displaystyle \tan\alpha$ and $\displaystyle \tan\beta$ are positive, $\displaystyle \tan\alpha\tan\beta$ is also positive.
So, $\displaystyle 1 - \tan\alpha\tan\beta $ is positive. and less than 1
And the minimum of $\displaystyle \tan\alpha + \tan\beta$ is zero, which occurs only when $\displaystyle \tan\alpha = 0$ or $\displaystyle \tan\beta = 0$, which is not the case.
Hence, we get
$\displaystyle 0 < \tan\alpha + \tan\beta < 1$
EDIT: A little too late...
Although you are given that both $\displaystyle 0<\alpha<\frac{{\pi}}{2}$ and $\displaystyle 0<\beta<\frac{{\pi}}{2}$
since $\displaystyle \alpha+\beta=\frac{{\pi}}{4}$
then both angles are in fact $\displaystyle <\frac{{\pi}}{4}$
hence $\displaystyle Tan\alpha<1$ and $\displaystyle Tan\beta<1$
Two positive fractions that are <1 when multiplied give a positive result <1
Hence, subtracting that result from 1 gives a value <1
What is not true here, is that you said that:
$\displaystyle \tan \alpha + \tan \beta = \tan45$
What is true is
$\displaystyle \dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}= 1$
Since $\displaystyle \tan \alpha + \tan \beta$ is divided by something, it is not equal to 1.