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Prove that, cos((3pi)/4+x) - cos((3pi)/4...

Prove that, `cos((3pi)/4+x) - cos((3pi)/4-x)=-sqrt(2)sinx`.

Text Solution

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LHS `=cos((3pi)/(4)+x)-cos((3pi)/(4)-x)`
`=1/2[2sin^(2)6x-2sin^(2)4x]`
`[therefore 2sin^(2)nx=(1-cos2nx)]`
`=1/2[cos8x-cos12x]`
`=1/2[cos8x-cos12x)]`
`1/2[2sin(8x+12x)/(2)sin(12x-8x)/(2)]`
`=sin10xsin2x=sin2xsin10x`= RHS
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