Question

Please solve this Question and please upload the steps.

Answer 1

Step-by-step explanation:

We have,

$\sin^2\bigg(\frac{A}{2}\bigg)+\sin^2\bigg(\frac{B}{2}\bigg)+\sin^2\bigg(\frac{C}{2}\bigg)$

$= \frac{1 - \cos(A) }{2}+ \frac{1 - \cos(B)}{2}+ \frac{1 - \cos(C)}{2}$

$= \frac{1}{2} - \frac{ \cos(A) }{2}+ \frac{1}{2} - \frac{\cos(B)}{2}+ \frac{1}{2} - \frac{\cos(C)}{2}$

$= \frac{3}{2} - \bigg \{ \frac{ \cos(A) }{2}+ \frac{\cos(B)}{2}+ \frac{\cos(C)}{2} \bigg \}$

$= \frac{3}{2} - \frac{1}{2} \bigg \{ \cos(A)+\cos(B)+ \cos(C) \bigg \}$

$= \frac{3}{2} - \frac{1}{2} \bigg \{ 2 \cos \bigg( \frac{A + B }{2}\bigg) \cos \bigg( \frac{A - B }{2}\bigg) +1 - 2\sin^{2} \bigg( \frac{C}{2} \bigg) \bigg \}$

$= \frac{3}{2} - \frac{1}{2} \bigg \{ 2 \cos \bigg( \frac{\pi - C}{2}\bigg) \cos \bigg( \frac{A - B }{2}\bigg) - 2\sin^{2} \bigg( \frac{C}{2} \bigg) \bigg \} - \frac{1}{2}$

$= 1 - \frac{1}{2} \bigg \{ 2 \cos \bigg( \frac{\pi}{2} - \frac{C}{2}\bigg) \cos \bigg( \frac{A - B }{2}\bigg) - 2\sin^{2} \bigg( \frac{C}{2} \bigg) \bigg \}$

$= 1 - \frac{1}{2} \bigg \{ 2 \sin\bigg( \frac{C}{2}\bigg) \cos \bigg( \frac{A - B }{2}\bigg) - 2\sin^{2} \bigg( \frac{C}{2} \bigg) \bigg \}$

$= 1 - \frac{1}{2} \bigg[ 2 \sin\bigg( \frac{C}{2}\bigg) \bigg \{ \cos \bigg( \frac{A - B }{2}\bigg) - \sin \bigg( \frac{C}{2} \bigg) \bigg \} \bigg]$

$= 1 - \frac{1}{2} \times 2 \bigg[ \sin\bigg( \frac{C}{2}\bigg) \bigg \{ \cos \bigg( \frac{A - B }{2}\bigg) - \sin \bigg( \frac{\pi - ( A + B )}{2} \bigg) \bigg \} \bigg]$

$= 1 - \bigg[ \sin\bigg( \frac{C}{2}\bigg) \bigg \{ \cos \bigg( \frac{A - B }{2}\bigg) - \sin \bigg( \frac{\pi}{2} - \frac{ A + B }{2} \bigg) \bigg \} \bigg]$

$= 1 - \bigg[ \sin\bigg( \frac{C}{2}\bigg) \bigg \{ \cos \bigg( \frac{A - B }{2}\bigg) - \cos \bigg( \frac{ A + B }{2} \bigg) \bigg \} \bigg]$

$= 1 - \bigg[ \sin\bigg( \frac{C}{2}\bigg) .2 \sin \bigg( \frac{A }{2}\bigg) \sin \bigg( \frac{ B }{2} \bigg) \bigg]$

$= 1 - \bigg[2 \sin \bigg( \frac{A }{2}\bigg) \sin \bigg( \frac{ B }{2} \bigg) \sin\bigg( \frac{C}{2}\bigg) \bigg]$

$= 1 - 2 \sin \bigg( \frac{A }{2}\bigg) \sin \bigg( \frac{ B }{2} \bigg) \sin\bigg( \frac{C}{2}\bigg)$