If `tan3A=cot(A-10^(@))` where 3A is an angle , find A.

Answer

To solve the equation \( \tan(3A) = \cot(A - 10^\circ) \), we can follow these steps:

### Step 1: Use the cotangent identity
We know that \( \cot(x) = \tan(90^\circ - x) \). Therefore, we can rewrite the equation as:
\[ \tan(3A) = \tan(90^\circ - (A - 10^\circ)) \]

### Step 2: Simplify the right-hand side
Now simplify the right-hand side:
\[ \tan(3A) = \tan(90^\circ - A + 10^\circ) = \tan(100^\circ - A) \]

### Step 3: Set the angles equal
Since the tangent function is periodic, we can set the angles equal to each other:
\[ 3A = 100^\circ - A + n \cdot 180^\circ \quad \text{(where \( n \) is an integer)} \]

For simplicity, we will consider \( n = 0 \):
\[ 3A = 100^\circ - A \]

### Step 4: Solve for \( A \)
Now, add \( A \) to both sides:
\[ 3A + A = 100^\circ \]
\[ 4A = 100^\circ \]
Now, divide by 4:
\[ A = \frac{100^\circ}{4} = 25^\circ \]

### Step 5: Conclusion
Thus, the value of \( A \) is:
\[ \boxed{25^\circ} \]