If tan x = cot (2x - 48), then find the value of x.

Options

A. `32^@`
B. `25^@`
C. `60^@`
D. `46^@`

Answer

To solve the equation \( \tan x = \cot(2x - 48) \), we can follow these steps:

### Step 1: Rewrite the cotangent in terms of tangent
We know that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, we can rewrite the equation:
\[ \tan x = \cot(2x - 48) = \frac{1}{\tan(2x - 48)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \tan x \cdot \tan(2x - 48) = 1 \]

### Step 3: Use the identity for tangent
We can also use the identity \( \tan A \tan B = 1 \) when \( A + B = 90^\circ \). This means:
\[ x + (2x - 48) = 90 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ x + 2x - 48 = 90 \]
\[ 3x - 48 = 90 \]

### Step 5: Solve for x
Add 48 to both sides:
\[ 3x = 90 + 48 \]
\[ 3x = 138 \]
Now, divide by 3:
\[ x = \frac{138}{3} = 46 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{46} \]