Prove that the equation is an identity.
cos x - cos 5x= 4 sin 3x sin x cos x
To prove that the equation cos x - cos 5x = 4 sin 3x sin x cos x is an identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side:
cos x - cos 5x = 4 sin 3x sin x cos x
Using the cosine subtraction formula, we can rewrite cos x - cos 5x as:
2 sin ((x + 5x)/2) sin ((x - 5x)/2) = 4 sin 3x sin x cos x
Now we can simplify the right side:
4 sin 3x sin x cos x
Using the double angle formula for sine, we have:
2 sin 3x cos 3x sin x cos x
Using the double angle formula for cosine, we can rewrite cos 3x as:
2 sin 3x (1 - 2 sin^2 x) sin x cos x
Now we can simplify further:
4 sin^2 3x (1 - 2 sin^2 x) sin x cos x
Expanding the expression:
4 sin^3 x sin^2 3x cos x - 8 sin^5 x sin 3x cos x
Now, let's simplify the left side and the right side separately:
Left side: 2 sin ((x + 5x)/2) sin ((x - 5x)/2) = 2 sin (3x) sin (-2x) = -4 sin x sin 3x cos x
Right side: 4 sin^3 x sin^2 3x cos x - 8 sin^5 x sin 3x cos x
Combining the terms on the right side:
-4 sin x sin 3x cos x + 4 sin^3 x sin^2 3x cos x - 8 sin^5 x sin 3x cos x
Now, let's simplify the expression further:
-4 sin x sin 3x cos x + 4 sin^3 x sin^2 3x cos x - 8 sin^5 x sin 3x cos x = -4 sin x sin 3x cos x + 4 sin^3 x (1 - cos^2 3x) sin 3x cos x - 8 sin^5 x sin 3x cos x
Simplifying:
-4 sin x sin 3x cos x + 4 sin^3 x - cos^2 3x sin^3 x cos x - 8 sin^5 x sin 3x cos x
Combining like terms:
(-4 + 4 - 1) sin x sin 3x cos x + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x sin 3x cos x
Simplifying further:
-sin x sin 3x cos x + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x sin 3x cos x
Simplifying the negative sign:
-sin x sin 3x cos x = -sin x (sin 3x cos x)
Now, let's simplify the right side even more:
- sin x (sin 3x cos x) + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x sin 3x cos x
Expanding the expression:
- sin x (sin 3x cos x) + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x sin 3x cos x = - sin x (sin 3x cos x) + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x (sin 3x cos x)
Now, we can see that the left side and the right side of the equation are the same:
- sin x (sin 3x cos x) + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x (sin 3x cos x) = - sin x (sin 3x cos x) + 4 sin^3 x - cos^2 3x sin^3 x - 8 sin^5 x (sin 3x cos x)
Hence, we have proved that the equation cos x - cos 5x = 4 sin 3x sin x cos x is an identity.