If α + β+ γ =180°, show that cos (α/2) + cos (β/2)+ cos (γ/2) = 4 cos ((β + γ)/4) cos ((γ + α)/4) cos ((α + β)/4).


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Question Number 57873 by 786786AM last updated on 13/Apr/19

$If α + β + γ = 180 °, show that \cos \frac{α}{2} + \cos \frac{β}{2} + \cos \frac{γ}{2} = 4 \cos \frac{β + γ}{4} \cos \frac{γ + α}{4} \cos \frac{α + β}{4} .$
Commented by tanmay.chaudhury50@gmail.com last updated on 13/Apr/19

$r e c h e c k t h e q u e s t i o n . . . p l s$
	Answered by math1967 last updated on 14/Apr/19

	$c o s \frac{α}{2} + c o s \frac{β}{2} + c o s \frac{γ}{2}$ $= 2 c o s \frac{α + β}{4} c o s \frac{α - β}{4} + c o s (\frac{180}{2} - \frac{α + β}{2})$ $= 2 c o s \frac{α + β}{4} c o s \frac{α - β}{4} + s i n \frac{α + β}{2}$ $= 2 c o s \frac{α + β}{4} c o s \frac{α - β}{4} + 2 s i n \frac{α + β}{4} c o s \frac{α + β}{4}$ $= 2 c o s \frac{α + β}{4} {c o s \frac{α - β}{4} + s i n \frac{α + β}{4}}$ $= 2 c o s \frac{α + β}{4} {c o s \frac{α - β}{4} + c o s (90 - \frac{α + β}{4}}$ $= 2 c o s \frac{α + β}{4} {2 c o s \frac{90 + \frac{α - β}{4} - \frac{α + β}{4}}{2} c o s \frac{90 - \frac{α + β}{4} - \frac{α - β}{4}}{2}$ $= 2 c o s \frac{α + β}{4} {2 c o s \frac{90 - \frac{2 β}{4}}{2} c o s \frac{90 - \frac{2 α}{4}}{2}}$ $= 4 c o s \frac{α + β}{4} c o s \frac{180 - β}{4} c o s \frac{180 - α}{4}$ $= 4 c o s \frac{α + β}{4} c o s \frac{α + γ}{4} c o s \frac{β + γ}{4} ★$ $★ (180 - α) = β + γ, (180 - β) = α + γ$
	Commented by 786786AM last updated on 14/Apr/19

	$Tk yu sir .$
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