If ` tan(cotx)=cot(tanx) `, then `sin2x ` is equal to

Answer

To solve the equation \( \tan(\cot x) = \cot(\tan x) \) and find the value of \( \sin 2x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \tan(\cot x) = \cot(\tan x) \]

### Step 2: Use the identity for cotangent
Recall that:
\[ \cot x = \frac{1}{\tan x} \]
Thus, we can rewrite \( \cot x \) in terms of \( \tan x \):
\[ \tan\left(\frac{1}{\tan x}\right) = \cot(\tan x) \]

### Step 3: Rewrite cotangent in terms of sine and cosine
Using the identity for cotangent:
\[ \cot y = \frac{\cos y}{\sin y} \]
we can rewrite the right-hand side:
\[ \cot(\tan x) = \frac{\cos(\tan x)}{\sin(\tan x)} \]

### Step 4: Set up the equation
Now we have:
\[ \tan\left(\frac{1}{\tan x}\right) = \frac{\cos(\tan x)}{\sin(\tan x)} \]

### Step 5: Use the identity for tangent
Recall that:
\[ \tan y = \frac{\sin y}{\cos y} \]
Thus, we can rewrite the left-hand side:
\[ \frac{\sin\left(\frac{1}{\tan x}\right)}{\cos\left(\frac{1}{\tan x}\right)} = \frac{\cos(\tan x)}{\sin(\tan x)} \]

### Step 6: Cross-multiply
Cross-multiplying gives us:
\[ \sin\left(\frac{1}{\tan x}\right) \sin(\tan x) = \cos\left(\frac{1}{\tan x}\right) \cos(\tan x) \]

### Step 7: Use the sine and cosine addition formula
This can be simplified using the sine and cosine addition formulas:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
and
\[ \cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)] \]

### Step 8: Simplify the equation
After applying the formulas, we can simplify the equation to find a relationship between \( x \) and \( n \):
\[ \sin 2x = \frac{4}{2n + 1} \quad \text{(where \( n \) is an integer)} \]

### Final Result
Thus, we find that:
\[ \sin 2x = \frac{4}{2n + 1} \]