`tan(5x-10^(2))=cot(5y+20^(@))` then value of (x+y) is

Options

A. `15^(@)`
B. `16^(@)`
C. `24^(@)`
D. `20^(@)`

Answer

To solve the equation \( \tan(5x - 10^\circ) = \cot(5y + 20^\circ) \), we can follow these steps:

### Step 1: Use the cotangent identity
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, we can rewrite the equation as:
\[ \tan(5x - 10^\circ) = \frac{1}{\tan(5y + 20^\circ)} \]
This implies:
\[ \tan(5x - 10^\circ) \tan(5y + 20^\circ) = 1 \]

### Step 2: Use the tangent identity
From the identity \( \tan(A) \tan(B) = 1 \), we know that:
\[ A + B = 90^\circ \]
Thus, we can set:
\[ (5x - 10^\circ) + (5y + 20^\circ) = 90^\circ \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 5x - 10^\circ + 5y + 20^\circ = 90^\circ \]
Combine like terms:
\[ 5x + 5y + 10^\circ = 90^\circ \]

### Step 4: Isolate \( x + y \)
Subtract \( 10^\circ \) from both sides:
\[ 5x + 5y = 80^\circ \]
Now, divide the entire equation by 5:
\[ x + y = \frac{80^\circ}{5} = 16^\circ \]

### Conclusion
Thus, the value of \( x + y \) is \( 16^\circ \).