What is the identity for cos3x?

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llltkl | College Teacher | (Level 3) Valedictorian

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cos3x=cos(2x+x)

Using the formula cos(A+B)=cosAcosB-sinAsinB

=cos2xcosx-sin2xsinx

Again,`cos2A=cos^2A-sin^2A=2cos^2A-1=1-2sin^2A`

and sin2A=2sinAcosA

Therefore,Cos3x=`(1-2sin^2x)cosx-(2sinxcosx)sinx`

`=cosx-2sin^2xcosx-2sin^2xcosx`

`=cosx-4sin^2xcosx`

`=cosx-4(1-cos^2x)cosx`

`=cosx-4cosx+4cos^3x`

`=4cos^3x-3cosx`

Hence the required identity for cos3x= `4cos^3x-3cosx`

 

 

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