If x= ` 20 ^(@) ` Evaluate :
` 12 sin ""(3x)/( 2) cos 3x + 5 tan (2x+ 5^(@)) - 3 cot ^(2)3x. `

Answer

To evaluate the expression given \( x = 20 \), we will follow these steps:

1. **Substitute x into the expression**:
\[ 12 \sin\left(\frac{3x}{2}\right) \cos(3x) + 5 \tan(2x + 5^\circ) - 3 \cot^2(3x) \]
Substituting \( x = 20 \):
\[ 12 \sin\left(\frac{3 \times 20}{2}\right) \cos(3 \times 20) + 5 \tan(2 \times 20 + 5^\circ) - 3 \cot^2(3 \times 20) \]
This simplifies to:
\[ 12 \sin(30) \cos(60) + 5 \tan(45^\circ) - 3 \cot^2(60) \]

2. **Calculate the trigonometric values**:
- \( \sin(30^\circ) = \frac{1}{2} \)
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \tan(45^\circ) = 1 \)
- \( \cot(60^\circ) = \frac{1}{\sqrt{3}} \)

3. **Substitute the trigonometric values back into the expression**:
\[ 12 \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) + 5(1) - 3 \left(\frac{1}{\sqrt{3}}\right)^2 \]

4. **Simplify each term**:
- First term:
\[ 12 \cdot \frac{1}{2} \cdot \frac{1}{2} = 12 \cdot \frac{1}{4} = 3 \]
- Second term:
\[ 5 \cdot 1 = 5 \]
- Third term:
\[ 3 \cdot \left(\frac{1}{3}\right) = 1 \]

5. **Combine the results**:
\[ 3 + 5 - 1 = 7 \]

Thus, the final answer is:
\[ \boxed{7} \]