Cos4x*cos2x=cos5x*cosx

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To prove that cos(4x)cos(2x) = cos(5x)cos(x), we can use trigonometric identities.

We have the following identities:

cos(4x) = cos(2x + 2x) = cos(2x)cos(2x) - sin(2x)sin(2x) = cos^2(2x) - sin^2(2x)

cos(5x) = cos(4x + x) = cos(4x)cos(x) - sin(4x)sin(x) = cos(4x)cos(x) - sin(4x)cos(π/2 - x)

Since sin(2x) = cos(4x)*cos(π/2 - 2x), we can rearrange cos(5x) as:

cos(5x) = cos(4x)cos(x) - cos(4x)cos(π/2 - 2x)

Now let's simplify the left side of the equation:

cos(4x)cos(2x) = (cos^2(2x) - sin^2(2x))cos(2x) = cos^2(2x)cos(2x) - sin^2(2x)cos(2x) = cos^2(2x)cos(2x) - (1 - cos^2(2x))cos(2x)

= cos^2(2x)cos(2x) - cos^2(2x)cos(2x) + cos^3(2x) = cos^3(2x)

Now, we can see that cos(5x)cos(x) = cos(4x)cos(x) - cos(4x)sin(x)*cos(π/2 - x)

= cos^3(2x)cos(x) - (cos^3(2x)cos(x) - cos^2(2x)*sin(x)sin(2x))

= cos^3(2x)cos(x) - cos^3(2x)cos(x) + cos^2(2x)*sin(x)cos(2x)

= cos^2(2x)*sin(x)cos(2x) = cos^3(2x)

Therefore, we have shown that cos(4x)cos(2x) = cos(5x)cos(x).