Solutions of triangles problem

Question

Prove that $$\left(a-b\right)^2\cos^2 \left(\frac{C}{2}\right) + \left(a+b\right)^2\sin^2\left(\frac{C}{2}\right)=c^2$$

My solution:-

I don't know what to do next..please give some hint. Or point out if I am wrong somewhere..

Answer 1

Using the Cosine, and "double angle" formulas we have:$$(a-b)^2\cos^2(\frac{C}{2})+(a+b)^2\sin^2(\frac{C}{2})=(a-b)^2\left(\dfrac{1+\cos C}{2}\right)+(a+b)^2\left(\dfrac{1-\cos C}{2}\right)=\dfrac{(a-b)^2+(a+b)^2}{2}+\dfrac{(a-b)^2-(a+b)^2}{2}\cdot \cos C=a^2+b^2-2ab\cos C = c^2$$

Answer 2

$$\text{L.H.S}=(a^2 +b^2)\left(\cos^2 \frac{C}{2} +\sin^2\frac{C}{2}\right)-2ab\left(\cos^2 \frac{C}{2} -\sin^2\frac{C}{2}\right)$$ $$=a^2 +b^2 - 2ab\cos C =c^2$$

Answer 3

By Mollweide’s Formulae

\begin{align*} \frac{a-b}{c} &= \frac{\sin \frac{A-B}{2}}{\cos \frac{C}{2}} \\ \frac{a+b}{c} &= \frac{\cos \frac{A-B}{2}}{\sin \frac{C}{2}} \\ \end{align*}

We have $$\left( \frac{a-b}{c} \right)^2 \cos^2 \frac{C}{2}+ \left( \frac{a+b}{c} \right)^2 \sin^2 \frac{C}{2}=1 $$