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You're right that division by zero is a problem if you want to use that identity. Rather than salvage that, I would recommend this identity instead:

$$
\arctan a + \arctan{a^{-1}} = \frac\pi2\qquad\text{for positive $a$}
$$

Proving this identity is easy: draw a $1 \times a$ rectangle with a diagonal and consider the two triangles thus formed.

Then show that $\frac{x}{x+1}$ is positive for all $x$ in the domain of $f$ and you're done.