<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] HINT:
Use Prosthaphaeresis Formulas
$$\cos2A+\cos2B+2\cos(A+B)=2\cos(A+B)[1+\cos(A-B)]$$
$$=4\cos(A+B)\cdot\sin^2\dfrac{A-B}2$$
Similarly, $$\sin2A+\sin2B+2\sin(A+B)=\cdots=4\sin(A+B)\cdot\sin^2\dfrac{A-B}2$$
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