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Question
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
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Solution
LHS = (cos α – cos β)2 + (sin α – sin β)2
`= (- 2 sin (alpha + beta)/2 sin (alpha - beta)/2)^2 + (2 cos (alpha + beta)/2 sin (alpha - beta)/2)^2`
`= 4 sin^2 (alpha + beta)/2 sin^2 (alpha - beta)/2 + 4 cos^2 (alpha + beta)/2 sin^2 (alpha - beta)/2`
`= 4 sin^2 (alpha - beta)/2 [sin^2 (alpha + beta)/2 + cos^2 (alpha + beta)/2]`
`= 4 sin^2 (alpha - beta)/2` = RHS
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