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]
