Prove that the equation is an identity.

cos x - cos 5x= 4 sin 3x sin x cos x

4sin3x sin x cos x

using double-angle formula,
2sin3x sin2x
using product-to-sum formula,
2(1/2 cos(3x-2x)-cos(3x+2x))
cos x - cos 5x

To prove that the equation cos x - cos 5x = 4 sin 3x sin x cos x is an identity, we need to show that it holds true for all values of x.

Using the identity for the cosine of the difference of two angles, we can rewrite the left side of the equation as:

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

Simplifying further:

= -2 sin(3x) sin(-2x)
= 2 sin(3x) sin(2x)

Now, let's factor the right side of the equation:

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

Comparing both sides of the equation, we can see that they are equal:

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

Hence, we have proven that the equation cos x - cos 5x = 4 sin 3x sin x cos x is an identity, since it holds true for all values of x.

To prove that an equation is an identity, we need to show that both sides of the equation are equivalent. In this case, we'll start with one side of the equation and try to simplify it to match the other side. Let's begin:

Start with the left-hand side (LHS) of the equation:
cos(x) - cos(5x)

To simplify this expression, we'll use the cosine angle subtraction identity:
cos(a) - cos(b) = -2 sin((a + b)/2) sin((a - b)/2)

Using this identity, we can rewrite the left-hand side of the equation:
LHS = -2 sin((x + 5x)/2) sin((x - 5x)/2)

Simplifying further:
LHS = -2 sin(6x/2) sin(-4x/2)
LHS = -2 sin(3x) sin(-2x)
LHS = 4 sin(3x) sin(x) [since sin(-2x) = -sin(2x)]

Now, let's compare the simplified left-hand side (LHS) with the right-hand side (RHS) of the equation:
4 sin(3x) sin(x) = 4 sin(3x) sin(x)

Since both sides are equal, we have shown that the equation cos(x) - cos(5x) = 4 sin(3x) sin(x) is indeed an identity.