Question:
If (1-sinA) (1-sinB) (1-sinC)=(1 sinA) (1 sin B)(1 sinC).....prove that each side is plus minus cos A cosB cosC
Answer:
Let (1 + sin a) * (1 + sin b) * (1 + sin c) = k = (1 - sin a) * (1 - sin b) * (1 - sin c)
So, (1 + sin a) * (1 + sin b) * (1 + sin c) = k .............1
and (1 - sin a) * (1 - sin b) * (1 - sin c) = k ......2
Now, multiply equation 1 and equation 2, we get
(1 + sin a) * (1 + sin b) * (1 + sin c) * (1 - sin a) * (1 - sin b) * (1 - sin c) = k
=> (1 - sin2 a) * (1 - sin2 b) * (1 - sin2 c) = k
=> cos2 a * cos2 b * cos2 c = k2
taking square root on both side, we get
=> ± cos a * cos b * cos c = k
So, (1 + sin a) * (1 + sin b) * (1 + sin c) = (1 - sin a) * (1 - sin b) * (1 - sin c) = ±cos a *cos b *cos c
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