cosA +cosB - cosC = -1 + 4cosA/2 cosB/2 sinC/2
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Step-by-step explanation:
We have to prove that :
cosa+cosb−cosc=4cos(a/2)cos(b/2)sin(c/2)−1
Here, we make an assumption that a+b+c=π
i.e, the sum of the angles a, b and c is 180°
L.H.S=(cosa+cosb)−cosc
=2cosa+b2cosa−b2−cosc
=2cos(π−c2)cos(a−b2)−cosc
=2cos(π2−c2)cos(a−b2)−cosc
=2sinc2cos(a−b2)−(1−2sin2c2)
=2sinc2(cosa−b2+sinc2)−1
=2sinc2(cosa−b2+sinπ−(a+b)2)−1
=2sinc2(cosa−b2+sin[π2−(a+b)2])−1
=2sinc2(cosa−b2+cos[(a+b)2])−1
=2sinc2(cos2a4+cos[(−2b)4])−1
=2sinc2(cosa2cos[−b2])−1
=4cos(a2)(cosb2)sin(c2)−1
=R.H.S
Happy math!
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