If cot 3x= sin ` 45^(@) cos 45 ^(@) + cos 60 ^(@) , ` find the value of x between ` 0^(@) and 90 ^(@) `

Answer

To solve the equation \( \cot 3x = \sin 45^\circ \cos 45^\circ + \cos 60^\circ \), we will follow these steps:

### Step 1: Calculate \( \sin 45^\circ \) and \( \cos 45^\circ \)
Both \( \sin 45^\circ \) and \( \cos 45^\circ \) are equal to \( \frac{1}{\sqrt{2}} \).

### Step 2: Substitute the values into the equation
Now, we substitute these values into the equation:
\[ \cot 3x = \sin 45^\circ \cos 45^\circ + \cos 60^\circ \]
This becomes:
\[ \cot 3x = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) + \cos 60^\circ \]

### Step 3: Calculate \( \cos 60^\circ \)
We know that \( \cos 60^\circ = \frac{1}{2} \).

### Step 4: Simplify the right-hand side
Now, we can simplify the right-hand side:
\[ \cot 3x = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 5: Solve for \( 3x \)
Since \( \cot 3x = 1 \), we can find \( 3x \):
\[ 3x = \cot^{-1}(1) \]
The angle whose cotangent is 1 is \( 45^\circ \):
\[ 3x = 45^\circ \]

### Step 6: Solve for \( x \)
Now, divide both sides by 3:
\[ x = \frac{45^\circ}{3} = 15^\circ \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{15^\circ} \]