In a , Prove that: ( A+B 2 )= C 2 - Vidyadip

Question

In a $\triangle\text{ABC},$ Prove that:
$\cos\Big(\frac{\text{A+B}}{2}\Big)=\sin\frac{\text{C}}{2}$

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Answer

We have $\text{A + B + C} = \pi$ $\big(\because$ sum of 3 angle of a triangle is $\pi=180^\circ\big)$
$\Rightarrow \text{A+B}={\pi}-\text{C}$
$$$\Rightarrow \frac{\text{A+B}}{2}=\frac{\pi-\text{C}}{2}$
$\Rightarrow \frac{\text{A+B}}{2}=\frac{\pi}{2}-\frac{\text{C}}{2}$
$\Rightarrow \cos=\Big(\frac{\text{A+B}}{2}\Big)=\cos\Big(\frac{\pi}{2}-\frac{C}{2}\Big)$
$\Rightarrow = \sin\frac{\text{C}}{2}$ $\Big(\because\cos\Big(\frac{\pi}{2}-\theta\Big)=\sin\theta\Big)$
Hence, $\cos\Big(\frac{\text{A+B}}{2}\Big)=\sin\frac{\text{C}}{2}$
$\text{Proved}$

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