Find the angle between the lines `y=(2-sqrt3)(x+5)` and `y=(2+sqrt3)(x-7)`.

Explanation

To find the angle between the lines given by the equations \( y = (2 - \sqrt{3})(x + 5) \) and \( y = (2 + \sqrt{3})(x - 7) \), we will follow these steps:

### Step 1: Identify the slopes of the lines
The equations of the lines can be rewritten in the slope-intercept form \( y = mx + c \), where \( m \) is the slope.

1. For the first line \( y = (2 - \sqrt{3})(x + 5) \):
- The slope \( m_1 = 2 - \sqrt{3} \).

2. For the second line \( y = (2 + \sqrt{3})(x - 7) \):
- The slope \( m_2 = 2 + \sqrt{3} \).

### Step 2: Use the formula for the angle between two lines
The formula to find the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \]

### Step 3: Substitute the values of \( m_1 \) and \( m_2 \)
Substituting the values we found:
\[ \tan \theta = \frac{(2 + \sqrt{3}) - (2 - \sqrt{3})}{1 + (2 - \sqrt{3})(2 + \sqrt{3})} \]

### Step 4: Simplify the numerator
The numerator simplifies as follows:
\[ (2 + \sqrt{3}) - (2 - \sqrt{3}) = \sqrt{3} + \sqrt{3} = 2\sqrt{3} \]

### Step 5: Simplify the denominator
Now, simplify the denominator:
\[ 1 + (2 - \sqrt{3})(2 + \sqrt{3}) = 1 + (4 - 3) = 1 + 1 = 2 \]

### Step 6: Combine the results
Now we can combine the results:
\[ \tan \theta = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Step 7: Find the angle \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
This corresponds to an angle of \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

### Final Answer
The angle between the lines is \( 60^\circ \).

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