Step-by-step Solution:

Correct Answer: C ($\theta = n\pi + \frac{\pi}{4}$)

To find the general solution of the equation, we can express it entirely in terms of a single trigonometric function, such as $\tan \theta$.

Step 1: Rewrite the Equation in terms of $\tan \theta$

The given equation is:

$$ \cot \theta + \tan \theta = 2 $$

Since $\cot \theta = \frac{1}{\tan \theta}$, we can substitute this into the equation:

$$ \frac{1}{\tan \theta} + \tan \theta = 2 $$

Step 2: Solve for $\tan \theta$

To eliminate the fraction, we multiply the entire equation by $\tan \theta$ (assuming $\tan \theta \neq 0$):

$$ 1 + \tan^2 \theta = 2 \tan \theta $$

Rearrange the terms to form a quadratic equation:

$$ \tan^2 \theta - 2 \tan \theta + 1 = 0 $$

This is a perfect square trinomial, which can be factored as:

$$ (\tan \theta - 1)^2 = 0 $$

Taking the square root of both sides gives:

$$ \tan \theta - 1 = 0 \implies \tan \theta = 1 $$

Step 3: Find the General Solution

We need to find the general solution for $\theta$ where $\tan \theta = 1$.

• The principal value of $\theta$ for which $\tan \theta = 1$ is $\frac{\pi}{4}$.
• The general solution for an equation of the form $\tan \theta = \tan \alpha$ is given by $\theta = n\pi + \alpha$, where $n$ is any integer.

Substituting our principal value, we get the final general solution:

$$ \theta = n\pi + \frac{\pi}{4} $$