`sin 67(1)/(2)^(@)+cos67(1)/(2)^(@)` is equal to

Options

A. `(1)/(2)sqrt(4+2sqrt(2))`
B. `(1)/(2)sqrt(4-2sqrt(2))`
C. `(1)/(4)(sqrt(4+2sqrt(2)))`
D. `(1)/(4)(sqrt(4-2sqrt(2)))`

Explanation

To solve the expression \( \sin 67.5^\circ + \cos 67.5^\circ \), we can use the following steps:

### Step 1: Recognize the Expression
We start with the expression:
\[ y = \sin 67.5^\circ + \cos 67.5^\circ \]

### Step 2: Square the Expression
Next, we square both sides:
\[ y^2 = (\sin 67.5^\circ + \cos 67.5^\circ)^2 \]

### Step 3: Expand the Square
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \), we expand:
\[ y^2 = \sin^2 67.5^\circ + \cos^2 67.5^\circ + 2 \sin 67.5^\circ \cos 67.5^\circ \]

### Step 4: Apply the Pythagorean Identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ y^2 = 1 + 2 \sin 67.5^\circ \cos 67.5^\circ \]

### Step 5: Use the Double Angle Identity
Recall that \( 2 \sin A \cos A = \sin(2A) \). Here, \( A = 67.5^\circ \):
\[ y^2 = 1 + \sin(135^\circ) \]

### Step 6: Evaluate \( \sin(135^\circ) \)
Since \( \sin(135^\circ) = \sin(180^\circ - 45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} \):
\[ y^2 = 1 + \frac{\sqrt{2}}{2} \]

### Step 7: Simplify the Expression
Now, we can write:
\[ y^2 = \frac{2}{2} + \frac{\sqrt{2}}{2} = \frac{2 + \sqrt{2}}{2} \]

### Step 8: Take the Square Root
Taking the square root of both sides gives:
\[ y = \sqrt{\frac{2 + \sqrt{2}}{2}} = \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2}} \]

### Step 9: Final Result
Thus, the final result for \( \sin 67.5^\circ + \cos 67.5^\circ \) is:
\[ \sin 67.5^\circ + \cos 67.5^\circ = \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2}} \]