If `cot(x+y)=1//sqrt(3), cot(x-y)=sqrt(3)` then what are the smallest positive values of x and y respectively ?

Options

A. `45^(@),30^(@)`
B. `30^(@),45^(@)`
C. `15,60^(@)`
D. `45^(@),15^(@)`

Explanation

To solve the problem, we start with the given equations:

1. \( \cot(x + y) = \frac{1}{\sqrt{3}} \)
2. \( \cot(x - y) = \sqrt{3} \)

### Step 1: Identify the angles from cotangent values
From the first equation, we know that:
\[ \cot(x + y) = \frac{1}{\sqrt{3}} \implies x + y = 60^\circ \]
This is because \( \cot(60^\circ) = \frac{1}{\sqrt{3}} \).

From the second equation, we have:
\[ \cot(x - y) = \sqrt{3} \implies x - y = 30^\circ \]
This is because \( \cot(30^\circ) = \sqrt{3} \).

### Step 2: Set up the system of equations
Now we have the following system of equations:
1. \( x + y = 60^\circ \) (Equation 1)
2. \( x - y = 30^\circ \) (Equation 2)

### Step 3: Solve the system of equations
To find \( x \) and \( y \), we can add both equations:
\[ (x + y) + (x - y) = 60^\circ + 30^\circ \]
This simplifies to:
\[ 2x = 90^\circ \implies x = \frac{90^\circ}{2} = 45^\circ \]

Now, we can substitute \( x \) back into one of the original equations to find \( y \). Using Equation 1:
\[ 45^\circ + y = 60^\circ \]
This gives:
\[ y = 60^\circ - 45^\circ = 15^\circ \]

### Final Answer
Thus, the smallest positive values of \( x \) and \( y \) are:
\[ x = 45^\circ, \quad y = 15^\circ \]

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