Prove the trigonometric identity:

tan(4x) = (4 tan(x) (1 - tan²(x))) / (1 - 6 tan²(x) + tan⁴(x))

To prove the trigonometric relation:

\[
\tan(4x) = \dfrac{{4\tan x(1 - \tan^2 x)}}{{1 - 6\tan^2 x + \tan^4 x}}
\]

### Step 1: Use the multiple angle identity for \(\tan(4x)\)

We know the identity for \(\tan(2A)\) is:

\[
\tan(2A) = \dfrac{2\tan(A)}{1 - \tan^2(A)}
\]

Now, apply this identity for \(\tan(4x)\) as \(\tan(2 \cdot 2x)\):

\[
\tan(4x) = \dfrac{2\tan(2x)}{1 - \tan^2(2x)}
\]

### Step 2: Find \(\tan(2x)\)

Using the identity for \(\tan(2A)\):

\[
\tan(2x) = \dfrac{2\tan(x)}{1 - \tan^2(x)}
\]

### Step 3: Substitute \(\tan(2x)\) into the expression for \(\tan(4x)\)

Now, substitute the expression for \(\tan(2x)\) into the formula for \(\tan(4x)\):

\[
\tan(4x) = \dfrac{2 \cdot \dfrac{2\tan(x)}{1 - \tan^2(x)}}{1 - \left( \dfrac{2\tan(x)}{1 - \tan^2(x)} \right)^2}
\]

### Step 4: Simplify the numerator

The numerator becomes:

\[
2 \cdot \dfrac{2\tan(x)}{1 - \tan^2(x)} = \dfrac{4\tan(x)}{1 - \tan^2(x)}
\]

### Step 5: Simplify the denominator

The denominator is:

\[
1 - \left( \dfrac{2\tan(x)}{1 - \tan^2(x)} \right)^2
\]

First, square the fraction:

\[
\left( \dfrac{2\tan(x)}{1 - \tan^2(x)} \right)^2 = \dfrac{4\tan^2(x)}{(1 - \tan^2(x))^2}
\]

So the denominator becomes:

\[
1 - \dfrac{4\tan^2(x)}{(1 - \tan^2(x))^2}
\]

Now, express 1 with a common denominator:

\[
1 = \dfrac{(1 - \tan^2(x))^2}{(1 - \tan^2(x))^2}
\]

So the denominator is:

\[
\dfrac{(1 - \tan^2(x))^2 - 4\tan^2(x)}{(1 - \tan^2(x))^2}
\]

Simplify the numerator:

\[
(1 - \tan^2(x))^2 - 4\tan^2(x) = 1 - 2\tan^2(x) + \tan^4(x) - 4\tan^2(x) = 1 - 6\tan^2(x) + \tan^4(x)
\]

So the denominator becomes:

\[
\dfrac{1 - 6\tan^2(x) + \tan^4(x)}{(1 - \tan^2(x))^2}
\]

### Step 6: Substitute the simplified expressions for the numerator and denominator

Now substitute the simplified numerator and denominator into the formula for \(\tan(4x)\):

\[
\tan(4x) = \dfrac{\dfrac{4\tan(x)}{1 - \tan^2(x)}}{\dfrac{1 - 6\tan^2(x) + \tan^4(x)}{(1 - \tan^2(x))^2}}
\]

### Step 7: Simplify the complex fraction

To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator:

\[
\tan(4x) = \dfrac{4\tan(x)}{1 - \tan^2(x)} \cdot \dfrac{(1 - \tan^2(x))^2}{1 - 6\tan^2(x) + \tan^4(x)}
\]

Cancel one factor of \(1 - \tan^2(x)\):

\[
\tan(4x) = \dfrac{4\tan(x)(1 - \tan^2(x))}{1 - 6\tan^2(x) + \tan^4(x)}
\]

### Final Answer:

Thus, we have proved that:

\[
\tan(4x) = \dfrac{4\tan(x)(1 - \tan^2(x))}{1 - 6\tan^2(x) + \tan^4(x)}
\]