If ` tan x = sin45^(@) cos 45^(@) + sin 30^(@)` then x is equal to

Options

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

Explanation

To solve the equation \( \tan x = \sin 45^\circ \cos 45^\circ + \sin 30^\circ \), we will follow these steps:

### Step 1: Substitute the known values of trigonometric functions
We know:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Substituting these values into the equation gives:
\[ \tan x = \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) + \frac{1}{2} \]

### Step 2: Simplify the right-hand side
Calculating \( \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \):
\[ \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \]
Thus, we have:
\[ \tan x = \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 3: Solve for \( x \)
Now, we need to find \( x \) such that:
\[ \tan x = 1 \]
The general solutions for \( \tan x = 1 \) are:
\[ x = 45^\circ + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]
For the principal value, we take:
\[ x = 45^\circ \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{45^\circ} \]

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