The Question reads - \(\displaystyle{\frac{{{\cos{{5}}}{x}+{\cos{{4}}}{x}}}{{{1}-{2}{\cos{{3}}}{x}}}}=-{\cos{{2}}}{x}-{\cos{{x}}}\)

Kennedy Ford

Kennedy Ford

Answered question

2022-04-09

The Question reads -
cos5x+cos4x12cos3x=cos2xcosx

Answer & Explanation

Videoad3u

Videoad3u

Beginner2022-04-10Added 15 answers

First I'll multiply through by (12cos(3x)) to do away with the negatives on the RHS and still have an equivalent equation. So it will suffice to prove that
cos5x+cos4x=(cos2x+cosx)(2cos3x1)
We will expand the RHS with complex exponentials.
P(cos2x+cosx)(2cos3x1)
=12(e2ix+e2ix+eix+eix)(e3ix+e3ix1)
=12(e5ix+eixe2ix+eix+e5ixe2ix+e4ix+e2ixeix+e2ix+e4ixeix)
=12(e5ix+e5ix+e4ix+e4ix)
=e5ix+e5ix2+e4ix+e4ix2
=cos(5x)+cos(4x)

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