`tantheta+cot theta=2`:

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. none of these

Answer

To solve the equation \( \tan \theta + \cot \theta = 2 \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, we can rewrite the equation as:
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

### Step 2: Let \( x = \tan \theta \)
Substituting \( x \) for \( \tan \theta \), we have:
\[ x + \frac{1}{x} = 2 \]

### Step 3: Multiply through by \( x \)
To eliminate the fraction, multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = 2x \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

### Step 5: Factor the quadratic
This can be factored as:
\[ (x - 1)^2 = 0 \]

### Step 6: Solve for \( x \)
Setting the factor to zero gives:
\[ x - 1 = 0 \implies x = 1 \]

### Step 7: Substitute back for \( \tan \theta \)
Since \( x = \tan \theta \), we have:
\[ \tan \theta = 1 \]

### Step 8: Find the angle \( \theta \)
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = 45^\circ \]

### Conclusion
Thus, the solution to the equation \( \tan \theta + \cot \theta = 2 \) is:
\[ \theta = 45^\circ \]