Understand the problem

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

Source of the problem
Vietnam National Mathematical Olympiad 1981
Topic
Trigonometry
Difficulty Level
Medium
Suggested Book
Challenge and Thrill of Pre-college Mathematics

Start with hints

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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