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Prove that 2cosx. Cos 3x- 2cos 5x. Cos 7...

Prove that `2cosx. Cos 3x- 2cos 5x. Cos 7x- cos 4x + cos12x=0`.

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LHS `=2cos x. cos 3x- 2cos5x. Cos7x - cos4x + cos 12x`
We know that, `2cosA.cosB=cos(A+B)+cos(A-B)`
`therefore 2cosx. Cos3x=cos 4x+cos 2x`
and `2cosx. Cos5x. Cos7x=cos 12x + cos 2x`
Now, LHS `=cos4x + cos2x - cos12x -cos2x-cos4x + cos12x=0` = RHS
Hence proved.
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