Proving Trigonometric Identities: tanA/(1-cotA) cotA/(1-tanA) secAcosecA 1

In this article, we will guide you through the process of proving the trigonometric identity:

(frac{tan A}{1 - cot A} cdot frac{cot A}{1 - tan A} sec A csc A 1)

Understanding the Trigonometric Functions

First, let's recall the basic definitions:

(tan A frac{sin A}{cos A}) and (cot A frac{cos A}{sin A}).

To prove the identity, we will start by substituting these definitions and simplifying the expression step by step.

Substitute the definitions of tangent and cotangent into the left-hand side:

(frac{frac{sin A}{cos A}}{1 - frac{cos A}{sin A}} cdot frac{frac{cos A}{sin A}}{1 - frac{sin A}{cos A}})

Now, let's simplify the denominators:

(1 - cot A 1 - frac{cos A}{sin A} frac{sin A - cos A}{sin A})

(1 - tan A 1 - frac{sin A}{cos A} frac{cos A - sin A}{cos A})

Substitute these into the fractions:

(frac{frac{sin A}{cos A}}{frac{sin A - cos A}{sin A}} cdot frac{frac{cos A}{sin A}}{frac{cos A - sin A}{cos A}})

Note that (cos A - sin A - (sin A - cos A)), so we can rewrite the second term:

(frac{sin^2 A}{cos A (sin A - cos A)} - frac{cos^2 A}{sin A (sin A - cos A)})

Combine these fractions:

(frac{sin^2 A sin A - cos^2 A cos A}{sin A cos A (sin A - cos A)})

Use the difference of cubes to factor the numerator:

(sin^3 A - cos^3 A (sin A - cos A) (sin^2 A sin A cos A cos^2 A))

So, the expression becomes:

(frac{(sin A - cos A) (sin^2 A sin A cos A cos^2 A)}{sin A cos A (sin A - cos A)} frac{sin^2 A sin A cos A cos^2 A}{sin A cos A})

Use the Pythagorean identity (sin^2 A cos^2 A 1):

(frac{1 sin A cos A}{sin A cos A})

Recognizing that (sec A frac{1}{cos A}) and (csc A frac{1}{sin A}), we can rewrite the expression as:

(frac{1}{sin A cos A} 1 sec A csc A 1)

This completes the proof that:

(frac{tan A}{1 - cot A} cdot frac{cot A}{1 - tan A} sec A csc A 1)