<s>[INST] prove that if $\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma=2$ then the triangle has a right angle prove that if $\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma=2$ then the triangle has a right angle. 
$\alpha,\beta,\gamma$ are the angles of the triangle.
I tried to use all kinds of trigonometric identities but it didn't work for me. it's to complex form me
Thanks. 
 [/INST] Let $\alpha$, $\beta$ and $\gamma$ be the angles of a triangle; then it holds
$$
\boxed{\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2+2\cos\alpha\cos\beta\cos\gamma}
$$
If this equals $2$, we conclude
$$
\cos\alpha\cos\beta\cos\gamma=0
$$
so one of the angles is a right angle.
Proof of the claim
Let's use that $\gamma=\pi-\alpha-\beta$, so $\cos\gamma=-\cos(\alpha+\beta)$. Then
\begin{align}
\cos^2\alpha+\cos^2\beta+\cos^2\gamma&=
\cos^2\alpha+\cos^2\beta+\cos^2(\alpha+\beta)\\
&=\cos^2\alpha+\cos^2\beta+\cos^2\alpha\cos^2\beta+\sin^2\alpha\sin^2\beta\\
&\qquad-2\sin\alpha\sin\beta\cos\alpha\cos\beta\\
&=\cos^2\alpha+\cos^2\beta+1-\cos^2\alpha-\cos^2\beta+2\cos^2\alpha\cos^2\beta\\
&\qquad-2\sin\alpha\sin\beta\cos\alpha\cos\beta\\
&=1+2\cos\alpha\cos\beta(\cos\alpha\cos\beta-\sin\alpha\sin\beta)\\
&=1-2\cos\alpha\cos\beta\cos\gamma
\end{align}
giving the final relation
$$
\boxed{\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1-2\cos\alpha\cos\beta\cos\gamma}
$$
Now
\begin{align}
\sin^2\alpha+\sin^2\beta+\sin^2\gamma
&=3-\cos^2\alpha-\cos^2\beta-\cos^2\gamma\\
&=2+2\cos\alpha\cos\beta\cos\gamma
\end{align}
as claimed at the beginning.
</s>