What is the value of `(sec45^0+1sqrt3)?`

Options

A. a. `(2+sqrt3)//2sqrt3`
B. b. `(2+2sqrt3)//sqrt3`
C. c. `(2+2sqrt3)//3`
D. d. `(sqrt6+1)//sqrt3`

Answer

To solve the expression \( \sec 45^\circ + \frac{1}{\sqrt{3}} \), we can follow these steps:

### Step 1: Find the value of \( \sec 45^\circ \)
The secant function is defined as the reciprocal of the cosine function. Therefore, we need to find \( \cos 45^\circ \).

\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

Thus,

\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \sqrt{2} \]

### Step 2: Substitute the value of \( \sec 45^\circ \) into the expression
Now that we have \( \sec 45^\circ = \sqrt{2} \), we can substitute this into the original expression:

\[ \sec 45^\circ + \frac{1}{\sqrt{3}} = \sqrt{2} + \frac{1}{\sqrt{3}} \]

### Step 3: Combine the terms
To combine \( \sqrt{2} \) and \( \frac{1}{\sqrt{3}} \), we need a common denominator. The common denominator for \( \sqrt{3} \) is \( \sqrt{3} \).

\[ \sqrt{2} = \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{\sqrt{3}} \]

Now we can rewrite the expression:

\[ \sqrt{2} + \frac{1}{\sqrt{3}} = \frac{\sqrt{6}}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{\sqrt{6} + 1}{\sqrt{3}} \]

### Step 4: Final answer
Thus, the value of \( \sec 45^\circ + \frac{1}{\sqrt{3}} \) is:

\[ \frac{\sqrt{6} + 1}{\sqrt{3}} \]