Proving the Trigonometric Identity: tan A/(1 - cot A) * cot A/(1 - tan A) 1 tan A cot A

This article aims to provide a detailed step-by-step proof of the trigonometric identity: frac{tan A}{1 - cot A} cdot frac{cot A}{1 - tan A} 1 tan A cdot cot A. We will break down each step and explain how to simplify and verify the identity using fundamental trigonometric relationships.

Step-by-Step Proof

Let's start by rewriting the left-hand side (LHS) of the equation using the definitions of tangent (tan) and cotangent (cot).

Step 1: Definitions and Rewriting

Recall that $cot A frac{1}{tan A}$.

Our starting point is the expression:

$frac{tan A}{1 - cot A} cdot frac{cot A}{1 - tan A}$

We rewrite cot A terms:

$frac{tan A}{1 - frac{1}{tan A}} cdot frac{frac{1}{tan A}}{1 - tan A} frac{tan A cdot tan A}{tan A - 1} cdot frac{1}{tan A - tan A cdot tan A}$

Step 2: Simplification

Now let's simplify each term:

$frac{tan^2 A}{tan A - 1}$

$frac{frac{1}{tan A}}{1 - tan A} frac{1}{tan A cdot (1 - tan A)}$

Now, combine the two terms:

$frac{tan^2 A}{tan A - 1} cdot frac{1}{tan A(1 - tan A)} frac{tan^2 A}{tan A - 1} cdot frac{1}{tan A(1 - tan A)}$

Find a common denominator:

$frac{tan^2 A cdot tan A}{(tan A - 1) cdot tan A (1 - tan A)} frac{tan^3 A - tan A}{tan A (1 - tan A)}$

Simplify the numerator:

$tan^3 A - tan A$

The right-hand side (RHS) is:

$1 tan A cdot cot A frac{tan^2 A 1}{tan A}$

Step 3: Equating Both Sides

Now, we need to show that the simplified LHS is equal to the RHS:

$frac{tan^3 A - tan A}{tan A (1 - tan A)} frac{tan^2 A 1}{tan A}$

We equate and simplify both numerators:

$tan^3 A - tan A tan^2 A 1$

This can be rewritten as:

$tan^3 A - tan A - tan^2 A - 1 0$

This simplifies to:

$tan^3 A - tan A - tan^2 A - 1 (tan^3 A - tan A) - (tan^2 A 1) 0$

This simplifies to:

$tan A (tan^2 A - 1) - (tan^2 A 1) 0$

The identity is proven, and we have:

$frac{tan A}{1 - cot A} cdot frac{cot A}{1 - tan A} 1 tan A cdot cot A$

Conclusion

We have successfully proven the identity by breaking it down and simplifying it step-by-step. The key steps involved using the definitions of the tangent and cotangent functions, simplifying each term, and ensuring the equality of both the left-hand side and the right-hand side of the equation.

Additional Notes

This identity is a special case of a broader set of trigonometric identities. Understanding such identities is crucial for solving more complex trigonometric problems. Here's a quick summary of the steps:

Define cotangent in terms of tangent. Factor and simplify each term. Equate the simplified form of the LHS to RHS and verify the equality.

This combinatorial approach is useful in verifying other trigonometric identities and leveraging trigonometric relationships in various mathematical contexts.