Find the general solution of :
`cotx+tanx=2cosecx`.

Answer

To find the general solution of the equation \( \cot x + \tan x = 2 \csc x \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting \( \cot x \) and \( \tan x \) in terms of sine and cosine:
\[ \cot x = \frac{\cos x}{\sin x}, \quad \tan x = \frac{\sin x}{\cos x} \]
Substituting these into the equation gives:
\[ \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = 2 \csc x \]

### Step 2: Combine the left-hand side
Next, we find a common denominator for the left-hand side:
\[ \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} = 2 \csc x \]
Using the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), we can simplify this to:
\[ \frac{1}{\sin x \cos x} = 2 \csc x \]

### Step 3: Rewrite \( \csc x \)
Recall that \( \csc x = \frac{1}{\sin x} \). Thus, we can rewrite the equation:
\[ \frac{1}{\sin x \cos x} = \frac{2}{\sin x} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ 1 = 2 \cos x \]

### Step 5: Solve for \( \cos x \)
From the equation \( 1 = 2 \cos x \), we can solve for \( \cos x \):
\[ \cos x = \frac{1}{2} \]

### Step 6: Find the general solution
The general solution for \( \cos x = \frac{1}{2} \) is:
\[ x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \]

Thus, the general solution is:
\[ x = 2n\pi + \frac{\pi}{3} \quad \text{or} \quad x = 2n\pi - \frac{\pi}{3}, \quad n \in \mathbb{Z} \]

### Final Answer
The general solution of the equation \( \cot x + \tan x = 2 \csc x \) is:
\[ x = 2n\pi \pm \frac{\pi}{3}, \quad n \in \mathbb{Z} \]

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