Prove that : `(1 - tan^(2) theta)/(1 + tan^(2) theta) = cos^(2) theta - sin^(2) theta`

Answer

To prove the identity

\[ \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos^2 \theta - \sin^2 \theta, \]

we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step 1: Rewrite \(\tan^2 \theta\)
Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, we can express \(\tan^2 \theta\) as:

\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}. \]

### Step 2: Substitute \(\tan^2 \theta\) in LHS
Substituting \(\tan^2 \theta\) into the LHS gives:

\[ \frac{1 - \frac{\sin^2 \theta}{\cos^2 \theta}}{1 + \frac{\sin^2 \theta}{\cos^2 \theta}}. \]

### Step 3: Simplify the fractions
To simplify the expression, we can multiply the numerator and the denominator by \(\cos^2 \theta\):

\[ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta + \sin^2 \theta}. \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can simplify the denominator:

\[ \cos^2 \theta + \sin^2 \theta = 1. \]

Thus, the expression simplifies to:

\[ \frac{\cos^2 \theta - \sin^2 \theta}{1} = \cos^2 \theta - \sin^2 \theta. \]

### Step 5: Conclusion
We have shown that:

\[ \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \cos^2 \theta - \sin^2 \theta, \]

which is the right-hand side (RHS) of the original equation. Therefore, we conclude that:

\[ \text{LHS} = \text{RHS}. \]

Hence, the identity is proved.