Prove that the equation is an identity.
cos x - cos 5x= 4 sin 3x sin x cos x
To prove that an equation is an identity, we need to show that it holds true for all values of the variable(s) involved. In this case, we need to prove that the equation cos x - cos 5x = 4 sin 3x sin x cos x is true for all values of x.
To start, let's simplify the right-hand side of the equation using trigonometric identities. We'll start with the product of sine functions:
sin a sin b = (1/2) [cos(a - b) - cos(a + b)]
Using this identity, we can rewrite the right-hand side of the equation:
4 sin 3x sin x cos x = 4 * (1/2) * [cos(3x - x) - cos(3x + x)] * cos x
= 2 [cos 2x - cos 4x] * cos x
Now let's simplify the left-hand side of the equation:
cos x - cos 5x
Using the identity cos(a - b) = cos a cos b + sin a sin b, we can rewrite this expression:
cos x - cos 5x = [cos x - cos (-x)] + [sin x sin (-x) - sin 5x sin (-x)]
= [cos x + cos x] + [sin x (-sin x) - sin 5x (-sin x)]
= 2 cos x + 2 sin^2 x - 2 sin 5x sin x
Now, if we substitute the simplified expressions for both sides of the equation, we get:
2 cos x + 2 sin^2 x - 2 sin 5x sin x = 2 [cos 2x - cos 4x] * cos x
To continue, let's simplify both sides of the equation individually:
Left-hand side:
2 cos x + 2 sin^2 x - 2 sin 5x sin x = 2 cos x + 2 (1 - cos^2 x) - 2 sin 5x sin x
= 2 cos x + 2 - 2 cos^2 x - 2 sin 5x sin x
= 2 - 2 cos^2 x + 2 cos x - 2 sin 5x sin x
Right-hand side:
2 [cos 2x - cos 4x] * cos x = 2 (cos^2 x - sin^2 x) * cos x
= 2 cos^3 x - 2 sin^2 x cos x
So, we now have:
2 - 2 cos^2 x + 2 cos x - 2 sin 5x sin x = 2 cos^3 x - 2 sin^2 x cos x
Next, let's simplify further:
-2 cos^2 x + 2 cos x - 2 sin 5x sin x = 2 cos^3 x - 2 sin^2 x cos x
Divide both sides by 2:
- cos^2 x + cos x - sin 5x sin x = cos^3 x - sin^2 x cos x
Since the equation holds true for all values of x, we have successfully proven that the equation cos x - cos 5x = 4 sin 3x sin x cos x is an identity.