Proving sin(a)-sin(b) = (a-b)cos(c) Using the Mean Value Theorem

What This Identity Actually Is

The formula sin(a) - sin(b) = (a - b)cos(c) looks simple. It is deceptively deep. Most people know the standard trig identity:

sin(a) - sin(b) = 2cos((a+b)/2) sin((a-b)/2)

This is useful. But it's not what we're proving here. The Mean Value Theorem version is different. It says there exists some value c between a and b where this holds exactly:

sin(a) - sin(b) = (a - b) · cos(c)

The c isn't a formula—it's a guarantee that such a point exists.

The Mean Value Theorem in 30 Seconds

If you need a refresher: the MVT states that for a function f continuous on [a,b] and differentiable on (a,b), there exists some c in (a,b) where:

f'(c) = (f(b) - f(a)) / (b - a)

Rearrange it:

f(b) - f(a) = f'(c) · (b - a)

That's all you need. No more theory. Let's apply it.

The Proof: Step by Step

Step 1: Identify Your Function

Pick f(x) = sin(x). This function is continuous everywhere and differentiable everywhere. It qualifies for the MVT without any restrictions.

Step 2: Apply the MVT Formula

Plug into f(b) - f(a) = f'(c) · (b - a):

sin(b) - sin(a) = cos(c) · (b - a)

Swap the sides to match the standard form:

sin(a) - sin(b) = (a - b)cos(c)

Done. That's the proof. One substitution.

Step 3: Understand What Just Happened

You didn't compute c. You didn't need to. The MVT guarantees c exists. It doesn't tell you where. That's the point—this is an existence proof, not a computation.

What Does the "c" Actually Mean?

The value c sits somewhere between a and b. It depends on a and b. You can't write c as a simple expression in terms of a and b alone.

Think of it this way: for any specific pair of numbers, there is some angle where the instantaneous rate of change of sine equals the average rate of change over that interval. That angle is c.

The practical implication: this identity tells you the difference in sine values is bounded by the maximum possible value of cosine. Since |cos(c)| ≤ 1:

|sin(a) - sin(b)| ≤ |a - b|

This is the Lipschitz continuity property of sine. The MVT gives it to you for free.

Comparing the Two Approaches

The trig identity is more useful for algebraic manipulation. The MVT version is more useful for proving inequalities and understanding function behavior.

How to Actually Use This

Here's a practical scenario: you need to bound |sin(x) - sin(y)| without computing sines.

By the MVT, there exists c between x and y where:

|sin(x) - sin(y)| = |x - y| · |cos(c)|

Since |cos(c)| ≤ 1:

|sin(x) - sin(y)| ≤ |x - y|

This works for any real numbers. No calculator needed. No approximation. Just the guarantee that cosine never exceeds magnitude 1.

Why This Proof Matters

You don't prove trig identities this way on a test. That's not the point. This proof shows:

• The MVT isn't abstract—it produces concrete results
• Existence theorems have practical numerical meaning
• The relationship between derivatives and function differences is direct

The formula sin(a) - sin(b) = (a - b)cos(c) is the MVT applied to sine. That's it. The elegance is in the simplicity.