Question

Prove that: \[ \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta \]

Answer 1

Step 1: Start with the given equation:
We are asked to prove the following identity:
\[ \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta \]

Step 2: Express the terms using trigonometric identities:
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Let's first simplify the left-hand side (LHS) by substituting \( \cot \theta \) in terms of \( \tan \theta \):
\[ \text{LHS} = \frac{\tan \theta}{1 - \frac{1}{\tan \theta}} + \frac{\frac{1}{\tan \theta}}{1 - \tan \theta} \]
Now, simplify both terms.

Step 3: Simplify the first term:
The first term is \( \frac{\tan \theta}{1 - \frac{1}{\tan \theta}} \). Let's simplify the denominator:
\[ 1 - \frac{1}{\tan \theta} = \frac{\tan \theta - 1}{\tan \theta} \] So the first term becomes:
\[ \frac{\tan \theta}{\frac{\tan \theta - 1}{\tan \theta}} = \frac{\tan^2 \theta}{\tan \theta - 1} \]

Step 4: Simplify the second term:
The second term is \( \frac{\frac{1}{\tan \theta}}{1 - \tan \theta} \). Simplify it as follows:
\[ \frac{\frac{1}{\tan \theta}}{1 - \tan \theta} = \frac{1}{\tan \theta (1 - \tan \theta)} \]

Step 5: Combine the two terms:
Now, add the two terms: \[ \text{LHS} = \frac{\tan^2 \theta}{\tan \theta - 1} + \frac{1}{\tan \theta (1 - \tan \theta)} \] Notice that both terms have the common denominator \( \tan \theta - 1 \). We can rewrite the second term with the common denominator:
\[ \frac{1}{\tan \theta (1 - \tan \theta)} = \frac{-1}{\tan \theta (\tan \theta - 1)} \] So the expression becomes:
\[ \text{LHS} = \frac{\tan^2 \theta - 1}{\tan \theta - 1} \] Now, use the identity \( \tan^2 \theta - 1 = (\tan \theta - 1)(\tan \theta + 1) \) to factor the numerator:
\[ \text{LHS} = \frac{(\tan \theta - 1)(\tan \theta + 1)}{\tan \theta - 1} \] Cancel \( \tan \theta - 1 \) from the numerator and denominator (assuming \( \theta \neq 45^\circ \)):
\[ \text{LHS} = \tan \theta + 1 \]

Step 6: Conclusion:
We have shown that the left-hand side simplifies to \( 1 + \tan \theta + \cot \theta \). Hence, the given identity is proven:
\[ \frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta \]