Answer :

GinnyAnswer
Let's simplify the given mathematical expression step-by-step:

Given expression:
[tex]\[ \frac{\sin 2\theta}{1 + \cos 2\theta} \times \frac{\cos \theta}{1 + \cos \theta} \][/tex]

We'll use trigonometric identities to simplify the expression.

Step 1: Simplify [tex]\(\frac{\sin 2\theta}{1 + \cos 2\theta}\)[/tex]

Recall the double-angle identities:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]

Also, use the identity:
[tex]\[ 1 + \cos 2\theta = 2 \cos^2 \theta \][/tex]

So our expression becomes:
[tex]\[ \frac{2 \sin \theta \cos \theta}{2 \cos^2 \theta} \][/tex]

Cancel out the 2 in the numerator and the denominator:
[tex]\[ \frac{\sin \theta}{\cos \theta} \][/tex]

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

Step 2: Insert back into the original expression

Now reinsert back into the original expression:
[tex]\[ \tan \theta \times \frac{\cos \theta}{1 + \cos \theta} \][/tex]

Step 3: Simplify [tex]\(\tan \theta \times \frac{\cos \theta}{1 + \cos \theta}\)[/tex]

We know:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]

So the expression becomes:
[tex]\[ \left( \frac{\sin \theta}{\cos \theta} \right) \times \frac{\cos \theta}{1 + \cos \theta} \][/tex]

Cancel out [tex]\(\cos \theta\)[/tex] in the numerator and the denominator:
[tex]\[ \frac{\sin \theta}{1 + \cos \theta} \][/tex]

Step 4: Use the half-angle identity

The half-angle identity states:
[tex]\[ \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \][/tex]

Thus we find:
[tex]\[ \frac{\sin \theta}{1 + \cos \theta} = \tan \frac{\theta}{2} \][/tex]

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

Hence, the given expression simplifies to:
[tex]\[ \tan \frac{\theta}{2} \][/tex]

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