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# Understand the problem

Prove that a triangle $ABC$ is right-angled if and only if
$$\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1$$

Trigonometry
Medium
##### Suggested Book
Challenge and Thrill of Pre-college Mathematics

Do you really need a hint? Try it first!

Familiarity with the trigonometric identities associated with a triangle is a must for any aspiring Olympian. Check the list given in the reference.
$ABC$ is right-angled iff $\cos A\cos B\cos C=0$.
Show that $\cos A\cos B\cos C=\frac{s^2-(2R+r)^2}{4R^2}$.

Combining hints 2 and 3, we see that $ABC$ is right-angled iff $s=2R+r$.

We know that $\sin A+\sin B+\sin C=\frac{s}{R}$ and $\cos A+\cos B+\cos C=1+\frac{r}{R}$,

hence $ABC$ is right-angled iff $\sin A+\sin B+\sin C=1+\cos A+\cos B+\cos C$.

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