<s>[INST] Prove identity in a triangle I want to show that if $ABC$ is a triangle then 
$$\sin^2(A/2)+ \sin^2(B/2) + \sin^2(C/2) =1-2\sin(A/2) \sin(B/2) \sin(C/2)$$
Well I eventually got it after much algebra, but I am looking for a shorter solution, or maybe even a geometric one? 
 [/INST] As $\dfrac A2+\dfrac B2=\dfrac\pi2-\dfrac C2,\sin\dfrac C2=\cos\dfrac{A+B}2$
Using Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$,
$$\sin^2\dfrac A2+\sin^2\dfrac B2+\sin^2\dfrac C2$$
$$=1-\left(\cos^2\dfrac A2-\sin^2\dfrac B2\right)+\sin^2\dfrac C2$$
$$=1-\cos\dfrac{A+B}2\cos\dfrac{A-B}2+\cos^2\dfrac{A+B}2$$
$$=1-\cos\dfrac{A+B}2\left(\cos\dfrac{A-B}2-\cos\dfrac{A+B}2\right)$$
Can you take it from here?
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