Solve for x :
`tan^(2)(x-5^(@))=3`

Answer

To solve the equation \( \tan^2(x - 5^\circ) = 3 \), we can follow these steps:

### Step 1: Take the square root of both sides
Starting with the equation:
\[ \tan^2(x - 5^\circ) = 3 \]
we take the square root of both sides:
\[ \tan(x - 5^\circ) = \sqrt{3} \quad \text{or} \quad \tan(x - 5^\circ) = -\sqrt{3} \]

### Step 2: Identify the angles for \(\tan\)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can set up the following equations:
1. \( x - 5^\circ = 60^\circ + n \cdot 180^\circ \) (for the positive root)
2. \( x - 5^\circ = -60^\circ + n \cdot 180^\circ \) (for the negative root)

where \( n \) is any integer.

### Step 3: Solve for \( x \) in both cases

**Case 1:**
\[ x - 5^\circ = 60^\circ + n \cdot 180^\circ \]
Adding \( 5^\circ \) to both sides:
\[ x = 65^\circ + n \cdot 180^\circ \]

**Case 2:**
\[ x - 5^\circ = -60^\circ + n \cdot 180^\circ \]
Adding \( 5^\circ \) to both sides:
\[ x = -55^\circ + n \cdot 180^\circ \]

### Step 4: Determine the general solution
The solutions for \( x \) can be expressed as:
1. \( x = 65^\circ + n \cdot 180^\circ \)
2. \( x = -55^\circ + n \cdot 180^\circ \)

### Step 5: Find specific solutions
For \( n = 0 \):
- From Case 1: \( x = 65^\circ \)
- From Case 2: \( x = -55^\circ \)

For \( n = 1 \):
- From Case 1: \( x = 65^\circ + 180^\circ = 245^\circ \)
- From Case 2: \( x = -55^\circ + 180^\circ = 125^\circ \)

### Final Answer
The general solutions for \( x \) are:
\[ x = 65^\circ + n \cdot 180^\circ \quad \text{and} \quad x = -55^\circ + n \cdot 180^\circ \]