Understanding Cotangent Subtraction Formulas: cot A - cot B ≠ cot(A - B)

Understanding the relationships between trigonometric functions is a fundamental aspect of mathematics. Specifically, the relationship between cot A - cot B and cot(A - B) is often misunderstood. In this article, we will explore the correct formulas and provide clarifications on why cot A - cot B ≠ cot(A - B).

Why cot A - cot B ≠ cot(A - B)

Let's start by recalling the correct cotangent subtraction formula. The formula for cot(A - B) is given by:

[ cot(A - B) frac{cot A cot B 1}{cot B - cot A}]

This formula shows that cot A - cot B and cot(A - B) are fundamentally different expressions. To further highlight this, let's derive and analyze the expressions for both.

Derivation of cot(A - B)

Starting with the definition of cotangent:

[ cot x frac{cos x}{sin x}]

Using this, we can express cot(A - B) as follows:

[ cot(A - B) frac{cos(A - B)}{sin(A - B)}]

Using the sum and difference identities:

[ cos(A - B) cos A cos B sin A sin B] [ sin(A - B) sin A cos B - sin B cos A]

Substituting these into the formula for cot(A - B):

[ cot(A - B) frac{cos A cos B sin A sin B}{sin A cos B - sin B cos A}]

Dividing numerator and denominator by sin A sin B to simplify:

[ cot(A - B) frac{cot A cot B 1}{cot B - cot A}]

Expression for cot A - cot B

Next, let's consider the expression cot A - cot B:

[ cot A - cot B frac{cos A}{sin A} - frac{cos B}{sin B}]

Combining the fractions under a common denominator:

[ cot A - cot B frac{sin B cos A - sin A cos B}{sin A sin B}]

Using the sine difference identity:

[ sin B cos A - sin A cos B sin(B - A)]

Thus:

[ cot A - cot B frac{sin(B - A)}{sin A sin B}]

Comparison of cot A - cot B and cot(A - B)

Comparing the expressions for cot A - cot B and cot(A - B) shows that they are indeed different:

[ cot A - cot B frac{sin(B - A)}{sin A sin B}] [ cot(A - B) frac{cot A cot B 1}{cot B - cot A}]

These expressions are not equal, hence demonstrating that:

[ cot A - cot B ≠ cot(A - B)]

Conclusion

Understanding the correct relationships between trigonometric functions is crucial for further mathematical analysis. As shown, the expression for cot A - cot B is fundamentally different from cot(A - B).

For further clarification or any questions about trigonometric identities, feel free to ask!