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Question

Find an expression for cos3x in terms of cosx.


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Solution

Solve for the value of cos3x in terms of cosx.

Expand 3xas 2x+x in the expression.

cos3x=cos2x+x

Use trignometric identity cos(a+b)=cosa.cosb-sina.sinb

Applying the identity in the above expression.

cos3x=cos2x.cosx-sin2x.sinx=-1+2cos2xcosx-2sinx.cosx.sinx[1+cos2x=2cos2x,sin2x=2sinxcosx]=2cos3x-2sin2xcosx-cosx=2cos3x-21-cos2xcosx-cosx=4cos3x-3cosx

Hence, cos3x=4cos3x-3cosx.


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