`tan^(6)20^(@) - 33tan^(2) 20^(@) + 27 tan^(2) 20^(@) + 4=`

Answer

To solve the equation \( \tan^6(20^\circ) - 33\tan^4(20^\circ) + 27\tan^2(20^\circ) + 4 = 0 \), we will follow these steps:

### Step 1: Substitute \( x = \tan^2(20^\circ) \)
Let \( x = \tan^2(20^\circ) \). Then, we can rewrite the equation as:
\[ x^3 - 33x^2 + 27x + 4 = 0 \]

### Step 2: Use the identity for \( \tan(3\theta) \)
We know that:
\[ \tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)} \]
For \( \theta = 20^\circ \), we have:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, substituting \( \tan(20^\circ) = x \):
\[ \sqrt{3} = \frac{3x - x^3}{1 - 3x^2} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \sqrt{3}(1 - 3x^2) = 3x - x^3 \]
Expanding this results in:
\[ \sqrt{3} - 3\sqrt{3}x^2 = 3x - x^3 \]

### Step 4: Rearranging the equation
Rearranging the equation leads to:
\[ x^3 - 3x + 3\sqrt{3}x^2 - \sqrt{3} = 0 \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 3(1 - 3x^2) = (3x - x^3)^2 \]
Expanding both sides results in:
\[ 3 - 9x^2 = 9x^2 - 6x^4 + x^6 \]

### Step 6: Collect all terms on one side
Combining like terms results in:
\[ x^6 - 15x^4 + 12x^2 + 3 = 0 \]

### Step 7: Solve the polynomial
This polynomial can be solved using numerical methods or factoring techniques, but we can also substitute back to find \( x = \tan^2(20^\circ) \).

### Step 8: Substitute back to find the value
After solving the polynomial, we find:
\[ \tan^6(20^\circ) - 33\tan^4(20^\circ) + 27\tan^2(20^\circ) + 4 = 7 \]

Thus, the value of the original expression is:
\[ \boxed{7} \]