What is the value of `(tan 45^@+1//sqrt2)`?

Options

A. `-sqrt3//2`
B. `(sqrt2+2)//2`
C. `(2sqrt2-sqrt3)//2`
D. `(sqrt2-1)//sqrt2`

Answer

To find the value of \( \tan 45^\circ + \frac{1}{\sqrt{2}} \), we can follow these steps:

### Step 1: Identify the value of \( \tan 45^\circ \)
We know that:
\[ \tan 45^\circ = 1 \]

### Step 2: Substitute the value of \( \tan 45^\circ \) into the expression
Now, we can substitute this value into the expression:
\[ \tan 45^\circ + \frac{1}{\sqrt{2}} = 1 + \frac{1}{\sqrt{2}} \]

### Step 3: Find a common denominator
To add these two terms, we need a common denominator. The common denominator here is \( \sqrt{2} \):
\[ 1 = \frac{\sqrt{2}}{\sqrt{2}} \]
Thus, we can rewrite the expression as:
\[ 1 + \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{\sqrt{2} + 1}{\sqrt{2}} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \):
\[ \frac{\sqrt{2} + 1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{2} + 1) \cdot \sqrt{2}}{2} \]
This simplifies to:
\[ \frac{2 + \sqrt{2}}{2} \]

### Final Result
Thus, the value of \( \tan 45^\circ + \frac{1}{\sqrt{2}} \) is:
\[ \frac{2 + \sqrt{2}}{2} \]