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Question

Prove that: cos5x+cos4x2cos3x1=cosx+cos2x.

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Solution

LHS =cos5x+cos4x12cos3x
cosC+cosD=2cos(C+D2)cos(CD2)
=⎢ ⎢ ⎢ ⎢2cos(9x2)cos(x2)12cos3x⎥ ⎥ ⎥ ⎥
Multiply and divide by cos(3x2)
=⎢ ⎢ ⎢ ⎢2cos(9x2)cos(x2)cos(3x2)cos(3x2)2cos3xcos3x2⎥ ⎥ ⎥ ⎥
=⎢ ⎢ ⎢2cos9x2×cosx2×cos3x2cos3x2cos9x2cos3x2⎥ ⎥ ⎥
2cosAcosB=cos(A+B)+cos(AB)
=⎢ ⎢ ⎢2cos9x2cosx2cos3x2cos9x2⎥ ⎥ ⎥=2cosx2cos3x2
=cos(2x)+cosx. RHS

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