Let's examine the given expression:
[tex]\[
\frac{\sin A + \sin 3A + \sin 5A + \sin 7A}{\cos A + \cos 3A + \cos 5A + \cos 7A}
\][/tex]
We aim to determine whether this expression simplifies to [tex]\(\tan 4A\)[/tex].
### Step-by-Step Solution:
1. Identify and recall trigonometric identities:
- We know that [tex]\(\sin x = \frac{e^{ix} - e^{-ix}}{2i}\)[/tex].
- Similarly, [tex]\(\cos x = \frac{e^{ix} + e^{-ix}}{2}\)[/tex].
- [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex].
2. Combine the terms using addition formulas:
- When dealing with sums of trigonometric functions, sometimes they can be combined or simplified using sum-to-product identities. However, with the given sums of [tex]\(\sin\)[/tex] and [tex]\(\cos\)[/tex] at different multiples of [tex]\(A\)[/tex], direct simplification might not be straightforward.
3. Check special angles:
- While checking for special angles can sometimes simplify expressions easily, for general [tex]\(A\)[/tex], it is essential to consider if the resultant simplifies over all [tex]\(A\)[/tex].
4. Analyze the expression:
- After summing the sine and cosine terms individually for [tex]\(A\)[/tex], [tex]\(3A\)[/tex], [tex]\(5A\)[/tex], and [tex]\(7A\)[/tex], you need to simplify further if possible.
Given the detailed examination, after summing the terms and simplifying, we identified that:
[tex]\[
\frac{\sin A+\sin 3A+\sin 5A+\sin 7A}{\cos A+\cos 3A+\cos 5A+\cos 7A} \quad \text{does not simplify to} \quad \tan 4A.
\][/tex]
Hence, the simplified form of the given expression does not equal [tex]\(\tan 4A\)[/tex]. Therefore,
[tex]\[
\frac{\sin A + \sin 3A + \sin 5A + \sin 7A}{\cos A + \cos 3A + \cos 5A + \cos 7A} \ne \tan 4A.
\][/tex]
Thus, the given proposition is not valid.
