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ત્રિકોણ $ABC$ માટે ,$\sin A + \sin B + \sin C = . . . .$
$4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$
$4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$
$4\cos \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$
$4\cos \frac{A}{2}\sin \frac{B}{2}\cos \frac{C}{2}$
Solution
(b) In $\Delta ABC,A + B + C = 180^\circ $
$ \Rightarrow \sin A + \sin B + \sin C $
$= 2\sin \frac{{A + B}}{2}\cos \frac{{A – B}}{2} + 2\sin \frac{C}{2}\cos \frac{C}{2}$
$ = 2\sin \left( {\frac{\pi }{2} – \frac{C}{2}} \right)\cos \frac{{A – B}}{2} + 2\cos \frac{C}{2}\sin \left( {\frac{\pi }{2} – \frac{{\overline {A + B} }}{2}} \right)$
$ = 2\cos \frac{C}{2}\cos \frac{{A – B}}{2} + 2\cos \frac{C}{2}\cos \frac{{A + B}}{2}$
$ = 2\cos \frac{C}{2}\left[ {\cos \frac{{A – B}}{2} + \cos \frac{{A + B}}{2}} \right]$
$ = 2\cos \frac{C}{2}\left( {2\cos \frac{A}{2}\cos \frac{B}{2}} \right) $
$= 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$ .
