Solve `(sin 10^(@))^(tan x+tan 3x)=tan 15^(@)+tan 30^(@)+ tan 15^(@). Tan 30^(@), x in (0, pi]`.

Explanation

To solve the equation \((\sin 10^\circ)^{(\tan x + \tan 3x)} = \tan 15^\circ + \tan 30^\circ + \tan 15^\circ\) for \(x\) in the interval \((0, \pi]\), we can follow these steps:

### Step 1: Simplify the Right-Hand Side (RHS)
First, we simplify the RHS:
\[ \tan 15^\circ + \tan 30^\circ + \tan 15^\circ = 2\tan 15^\circ + \tan 30^\circ \]
Using known values:
\[ \tan 15^\circ = 2 - \sqrt{3}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus,
\[ 2\tan 15^\circ = 2(2 - \sqrt{3}) = 4 - 2\sqrt{3} \]
So,
\[ RHS = 4 - 2\sqrt{3} + \frac{1}{\sqrt{3}} \]

### Step 2: Evaluate the Right-Hand Side
Calculating \(4 - 2\sqrt{3} + \frac{1}{\sqrt{3}}\):
To combine these terms, we can convert \(4\) into a fraction:
\[ 4 = \frac{4\sqrt{3}}{\sqrt{3}} \Rightarrow RHS = \frac{4\sqrt{3}}{\sqrt{3}} - \frac{2\sqrt{3}\cdot\sqrt{3}}{3} + \frac{1}{\sqrt{3}} = \frac{4\sqrt{3} - 6 + 1}{\sqrt{3}} = \frac{4\sqrt{3} - 5}{\sqrt{3}} \]

### Step 3: Set Up the Equation
Now we have:
\[ (\sin 10^\circ)^{(\tan x + \tan 3x)} = \frac{4\sqrt{3} - 5}{\sqrt{3}} \]

### Step 4: Analyze the Left-Hand Side (LHS)
For the LHS to equal \(1\), we need:
\[ \tan x + \tan 3x = 0 \]
This implies:
\[ \tan 3x = -\tan x \]

### Step 5: Use the Tangent Addition Formula
Using the tangent addition formula:
\[ \tan(3x + x) = \frac{\tan 3x + \tan x}{1 - \tan 3x \tan x} = 0 \]
This gives us:
\[ 3x + x = n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus:
\[ 4x = n\pi \Rightarrow x = \frac{n\pi}{4} \]

### Step 6: Find Solutions in the Interval
Now we find \(x\) in the interval \((0, \pi]\):
- For \(n = 1\): \(x = \frac{\pi}{4}\)
- For \(n = 2\): \(x = \frac{2\pi}{4} = \frac{\pi}{2}\)
- For \(n = 3\): \(x = \frac{3\pi}{4}\)
- For \(n = 4\): \(x = \frac{4\pi}{4} = \pi\)

Thus, the solutions are:
\[ x = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \]

### Final Solution
The final solutions for the equation are:
\[ x = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \]