Answer :

Certainly! Let's solve the equation step by step using trigonometric identities and simplifications.

Given the equation:

[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]

### Step-by-Step Solution:

#### 1. Simplify the Left-Hand Side (LHS):

The left-hand side of the equation is:

[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} \][/tex]

We use the double angle identities:

[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]

Substitute these into the LHS:

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

#### 2. Simplify the Right-Hand Side (RHS):

The right-hand side of the equation is:

[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]

Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]. Substitute this into the RHS:

[tex]\[ \frac{1 - \frac{\sin \theta}{\cos \theta}}{1 + \frac{\sin \theta}{\cos \theta}} \][/tex]

Combine the fractions:

[tex]\[ \frac{\frac{\cos \theta - \sin \theta}{\cos \theta}}{\frac{\cos \theta + \sin \theta}{\cos \theta}} \][/tex]

Simplify the expression by multiplying both the numerator and the denominator by [tex]\(\cos \theta\)[/tex]:

[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

So the RHS is:

[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

### 3. Compare the Simplified LHS and RHS:

We now have:

[tex]\[ \frac{\cos^2 \theta - \sin^2 \theta}{1 + 2 \sin \theta \cos \theta} \quad \text{and} \quad \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

Next, we observe that:

[tex]\[ \cos^2 \theta - \sin^2 \theta = (\cos \theta - \sin \theta)(\cos \theta + \sin \theta) \][/tex]
[tex]\[ 1 + 2 \sin \theta \cos \theta = (\cos \theta + \sin \theta)^2 \][/tex]

So we substitute back into the LHS:

[tex]\[ \frac{(\cos \theta - \sin \theta)(\cos \theta + \sin \theta)}{(\cos \theta + \sin \theta)^2} \][/tex]

Simplify the fraction by canceling [tex]\((\cos \theta + \sin \theta)\)[/tex] in the numerator and the denominator:

[tex]\[ \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

### 4. Conclusion:

Both the simplified left-hand side and right-hand side expressions match:

[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

[tex]\[ \frac{1 - \tan \theta}{1 + \tan \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta} \][/tex]

Therefore, we have verified that:

[tex]\[ \frac{\cos 2\theta}{1 + \sin 2\theta} = \frac{1 - \tan \theta}{1 + \tan \theta} \][/tex]

is indeed true.

Therefore, the equation holds.