prove the identity :
cos3x =4 cos^3x-3cosx
here is one version
https://socratic.org/questions/how-do-you-solve-the-identity-cos3x-4cos-3x-3cosx
To prove the identity cos3x = 4cos^3x - 3cosx, we need to bring both sides of the equation to the same form using trigonometric identities.
Let's start by expanding the right side of the equation: 4cos^3x - 3cosx.
Using the trigonometric identity cos^2x = 1 - sin^2x, we can rewrite the equation as:
4(1 - sin^2x)cosx - 3cosx
Expanding further:
4cosx - 4sin^2x * cosx - 3cosx
Combining like terms:
(cosx) * (4 - 4sin^2x - 3)
Simplifying:
(cosx) * (1 - 4sin^2x)
Now, let's focus on the left side of the equation, cos3x.
Using the triple angle identity cos(3x) = 4cos^3x - 3cosx, we can substitute cos3x with 4cos^3x - 3cosx:
4cos^3x - 3cosx = 4cos^3x - 3cosx
We have now shown that both sides of the equation are equivalent, and thus, we have proved the identity:
cos3x = 4cos^3x - 3cosx