What is the value of `(2cot 45^@ + sqrt(3) /sqrt(2))` ?

Options

A. A) `5//6`
B. B) `(4 + sqrt()3)//2sqrt()3`
C. C) `(2sqrt()2 + sqrt()3)//sqrt2`
D. D) `3sqrt()3//2`

Answer

To find the value of \(2 \cot 45^\circ + \frac{\sqrt{3}}{\sqrt{2}}\), we can follow these steps:

### Step 1: Calculate \( \cot 45^\circ \)
The cotangent of \(45^\circ\) is defined as:
\[ \cot 45^\circ = \frac{1}{\tan 45^\circ} = 1 \]

### Step 2: Multiply by 2
Now, we multiply the value of \( \cot 45^\circ \) by 2:
\[ 2 \cot 45^\circ = 2 \times 1 = 2 \]

### Step 3: Add \( \frac{\sqrt{3}}{\sqrt{2}} \)
Next, we add \(2\) to \( \frac{\sqrt{3}}{\sqrt{2}} \):
\[ 2 + \frac{\sqrt{3}}{\sqrt{2}} \]

### Step 4: Find a common denominator
To add these two terms, we need a common denominator. The common denominator here is \( \sqrt{2} \):
\[ 2 = \frac{2\sqrt{2}}{\sqrt{2}} \]
Now we can rewrite the expression:
\[ \frac{2\sqrt{2}}{\sqrt{2}} + \frac{\sqrt{3}}{\sqrt{2}} = \frac{2\sqrt{2} + \sqrt{3}}{\sqrt{2}} \]

### Final Answer
Thus, the value of \(2 \cot 45^\circ + \frac{\sqrt{3}}{\sqrt{2}}\) is:
\[ \frac{2\sqrt{2} + \sqrt{3}}{\sqrt{2}} \]

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