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Question
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
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Solution
L.H.S. = cos `((3x)/4 + x) - cos ( (3x) /4 - x)`
= - 2sin `[((3x)/4 + x) /2+ ( (3x) /4 - x)/2] sin [((3pi)/4)/2- ((3pi)/4-x)/2]`
[cos (a + b) - cos (a - b) = - 2sin `((a +b)/2) sin ((a + b )/2)`
= - 2sin `((3pi)/4) sin x = - 2 (pi - pi/4) sin x`
= - 2 x `1/sqrt2 sin x = - sqrt2 sin x` = R.H.S.
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