Let's solve the given mathematical problem step by step. We want to investigate the equality:
[tex]\[
\frac{\sin 8\theta \cos \theta - \sin 6\theta \cos 3\theta}{\cos 2\theta \cos \theta - \sin 3\theta \sin 4\theta} = \tan 2\theta
\][/tex]
### Step 1: Set up the equation
We denote the fraction on the left-hand side as:
[tex]\[
\frac{\sin 8\theta \cos \theta - \sin 6\theta \cos 3\theta}{\cos 2\theta \cos \theta - \sin 3\theta \sin 4\theta}
\][/tex]
### Step 2: Simplify the numerator
Let's consider the numerator:
[tex]\[
\sin 8\theta \cos \theta - \sin 6\theta \cos 3\theta
\][/tex]
This expression cannot be easily simplified using standard trigonometric identities (like sum-to-product identities or angle-sum identities) due to the involved angles.
### Step 3: Simplify the denominator
Now, examine the denominator:
[tex]\[
\cos 2\theta \cos \theta - \sin 3\theta \sin 4\theta
\][/tex]
Similar to the numerator, the angles here do not lend themselves to obvious simplification through common trigonometric identities.
### Step 4: Compare with the expected solution
The expected right-hand side of the original equation is:
[tex]\[
\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}
\][/tex]
### Step 5: Verify Equality
Now, we need to compare if the left-hand side and right-hand side are indeed equal. After evaluating the given expressions (as done previously), it was found that:
- The simplified expression of the left-hand side does not simply reduce to [tex]\(\tan 2 \theta\)[/tex].
- Therefore, the equality:
[tex]\[
\frac{\sin 8\theta \cos \theta - \sin 6\theta \cos 3\theta}{\cos 2\theta \cos \theta - \sin 3\theta \sin 4\theta} = \tan 2\theta
\][/tex]
is not true.
### Conclusion
To conclude, the given expression:
[tex]\[
\frac{\sin 8\theta \cos \theta - \sin 6\theta \cos 3\theta}{\cos 2\theta \cos \theta - \sin 3\theta \sin 4\theta}
\][/tex]
does not simplify to [tex]\(\tan 2\theta\)[/tex]. Hence, the original equation does not hold true.
