<s>[INST] Finding the hypotenuse ( Trigonometric Identities!) If the sides of a right-angled  triangle are $\cos 2a + \cos 2b + 2\cos(a+b)$ and $\sin 2a + \sin 2b + 2\sin(a+b)$, find the hypotenuse-
I can simplify this but I end up having a lot of terms in the end :/ 
 [/INST] $$\cos 2a+\cos 2b+2\cos(a+b)=\cos^2 a-\sin^2 a + \cos^2b-\sin^2b+2\cos a\cos b-2\sin a\sin b=(\cos a+\cos b)^2-(\sin a+\sin b)^2$$
$$\sin2a+\sin2b+2\sin(a+b)=2\sin a\cos a+2\sin b\cos b+2\sin a\cos b+2\cos a\sin b=2(\sin a+\sin b)(\cos a+\cos b)$$
Let $\cos a+\cos b=u$, $\sin a+\sin b=v$.
$$\text{Hypotenuse}=\sqrt{(u^2-v^2)^2+(2uv)^2}=u^2+v^2=(\cos a+\cos b)^2+(\sin a+\sin b)^2=2(1+\cos(a-b))$$
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