Computing sum of lengths of legs of a right triangle

Question

Consider a right triangle $\triangle ABC$, where the right angle is $\angle A\hat CB$, as in the picture below. Let $\alpha=\angle A\hat BC$.

Problem: To determine the sum of the lengths of the legs of $\triangle ABC$ as a function of $\alpha$ and of the radius $r$ of the incircle of $\triangle ABC$.

Actually, the problem is not just this. If it was just this, then my answer would be$$r\left(2+\cot\left(\frac\alpha2\right)+\cot\left(\frac\pi4-\frac\alpha2\right)\right).\tag1$$

However, this is a problem from a textbook on plane Geometry and Trigonometry and the given answer is$$r\left(2+\frac{1+\sin(\alpha)+\cos(\alpha)}{\sin(\alpha)\cos(\alpha)}\right).\tag2$$Of course, it is easy to show that $(1)\iff(2)$, but my guess is that whoever made this exercise meant to find $(2)$ directly from the picture. Any idea about how to do that?

Answer 1

See diagram:

$$ {r\over\cos\alpha}+r\tan\alpha+r+r+r\cot\alpha+{r\over\sin\alpha}= r\left(2+{1+\sin\alpha+\cos\alpha\over\sin\alpha\cos\alpha}\right) $$

Answer 2

1 and 2 are not equivalent. Consider substituting $\alpha = \pi/4$.

https://www.wolframalpha.com/input/?i=2%2Bcot(pi%2F8)%2Bcot(pi%2F4-pi%2F8)

https://www.wolframalpha.com/input/?i=2%2B(1%2Bsin(pi%2F4)cos(pi%2F4))%2F(sin(pi%2F4)cos(pi%2F4))