Question

sina + sinß + siny - sin(a + B + y) = 4 sin(a+b /2). sin (b+y/2 ).sin (y+a/2).​

Answer 1

Step-by-step explanation:

We have,

$\tt{ \green{sin( \alpha) + sin(\beta) + sin( \gamma ) - \sin( \alpha + \beta + \gamma ) } }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: cos \bigg( \dfrac{ \alpha - \beta }{2}\bigg) + 2 \: cos \bigg( \dfrac{\gamma + \alpha + \beta + \gamma }{2}\bigg) \: sin \bigg( \dfrac{\gamma - \alpha - \beta - \gamma }{2}\bigg) }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: cos \bigg( \dfrac{ \alpha - \beta }{2}\bigg) + 2 \: cos \bigg( \dfrac{ \alpha + \beta + 2\gamma }{2}\bigg) \: sin \bigg( \dfrac{ - \alpha - \beta }{2}\bigg) }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: cos \bigg( \dfrac{ \alpha - \beta }{2}\bigg) - 2 \: cos \bigg( \dfrac{ \alpha + \beta + 2\gamma }{2}\bigg) \: sin \bigg( \dfrac{ \alpha + \beta }{2}\bigg) }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg \{ cos \bigg( \dfrac{ \alpha - \beta }{2}\bigg) - cos \bigg( \dfrac{ \alpha + \beta + 2\gamma }{2}\bigg) \bigg \} }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[ - 2 \:sin \bigg \{ \dfrac{1}{2} \bigg(\dfrac{ \alpha - \beta }{2} + \dfrac{ \alpha + \beta + 2\gamma }{2}\bigg) \bigg \} \: sin \bigg \{ \dfrac{1}{2} \bigg(\dfrac{ \alpha - \beta }{2} - \dfrac{ \alpha + \beta + 2\gamma }{2}\bigg) \bigg \} \bigg ] }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[ - 2 \:sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ \alpha - \beta + \alpha + \beta + 2\gamma }{2}\bigg) \bigg \} \: sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ \alpha - \beta - \alpha - \beta - 2\gamma }{2}\bigg) \bigg \} \bigg ] }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[ - 2 \:sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ \alpha + \alpha + 2\gamma }{2}\bigg) \bigg \} \: sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ - \beta - \beta - 2\gamma }{2}\bigg) \bigg \} \bigg ] }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[ - 2 \:sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ 2 \alpha + 2\gamma }{2}\bigg) \bigg \} \: sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ -2 \beta - 2\gamma }{2}\bigg) \bigg \} \bigg ] }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[2 \:sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ 2 \alpha + 2\gamma }{2}\bigg) \bigg \} \: sin \bigg \{ \dfrac{1}{2} \bigg( \dfrac{ 2 \beta + 2\gamma }{2}\bigg) \bigg \} \bigg ] }$

$\sf{ = 2 \: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: \bigg[2 \:sin \bigg( \dfrac{ \gamma + \alpha }{2}\bigg) \: sin \bigg( \dfrac{ \beta + \gamma }{2}\bigg) \bigg ] }$

$\sf{ = 4\: sin \bigg( \dfrac{ \alpha +\beta }{2}\bigg) \: sin \bigg( \dfrac{ \beta + \gamma }{2}\bigg) \:sin \bigg( \dfrac{ \gamma + \alpha }{2}\bigg) }$