 tanA/1-cotA + cotA/1-tanA = 1+ tanA + cotA

We are given the equation to prove:

\[
\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = 1 + \tan A + \cot A
\]

### Steps to Prove the Identity:

1. **Convert \(\cot A\) to \(\tan A\)**:
   We know that \(\cot A = \frac{1}{\tan A}\). Substituting this into the equation:

   \[
   \frac{\tan A}{1 - \frac{1}{\tan A}} + \frac{\frac{1}{\tan A}}{1 - \tan A}
   \]

2. **Simplify the terms**:
   Simplify the denominator in the first fraction:

   \[
   \frac{\tan A}{\frac{\tan A - 1}{\tan A}} = \frac{\tan A \cdot \tan A}{\tan A - 1} = \frac{\tan^2 A}{\tan A - 1}
   \]
   
   Simplify the second fraction:
   
   \[
   \frac{\frac{1}{\tan A}}{1 - \tan A} = \frac{1}{\tan A (1 - \tan A)}
   \]

   Now the equation becomes:

   \[
   \frac{\tan^2 A}{\tan A - 1} + \frac{1}{\tan A (1 - \tan A)}
   \]

3. **Get common denominators**:
   The common denominator of these two fractions will be \((\tan A - 1)(1 - \tan A)\). So we rewrite the fractions:

   \[
   \frac{\tan^2 A}{\tan A - 1} + \frac{1}{1 - \tan A}
   \]

4. **Combine fractions**:
   Add the two fractions with the common denominator:

   \[
   \frac{\tan^2 A + 1}{(1 - \tan A)(\tan A - 1)}
   \]

   Since \((1 - \tan A)(\tan A - 1) = -(\tan A - 1)^2\), the equation becomes:

   \[
   \frac{\tan^2 A + 1}{-(\tan A - 1)^2}
   \]

5. **Use trigonometric identities**:
   Now use the identity \( \tan^2 A + 1 = \sec^2 A\). So the expression simplifies to:

   \[
   \frac{\sec^2 A}{-(\tan A - 1)^2}
   \]

   This should simplify further to give us the right-hand side, but it appears there's a sign issue in how we've proceeded. Let me review the algebra and approach it differently

Explore how to prove the trigonometric identity tanA/(1 - cotA) + cotA/(1 - tanA) = 1 + tanA + cotA step-by-step. This problem involves applying trigonometric identities and simplification techniques. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation