The value of cos `105^@ + sin 105^@` is

Options

A. 0
B. `(sqrt3//2)`
C. `1//sqrt2`
D. `(sqrt3+1)//2`

Answer

To find the value of \( \cos 105^\circ + \sin 105^\circ \), we can follow these steps:

### Step 1: Rewrite the angles
We can express \( 105^\circ \) as \( 90^\circ + 15^\circ \).

### Step 2: Use trigonometric identities
Using the angle addition formulas:
- \( \cos(90^\circ + \theta) = -\sin(\theta) \)
- \( \sin(90^\circ + \theta) = \cos(\theta) \)

Thus, we have:
\[ \cos 105^\circ = \cos(90^\circ + 15^\circ) = -\sin 15^\circ \]
\[ \sin 105^\circ = \sin(90^\circ + 15^\circ) = \cos 15^\circ \]

### Step 3: Substitute back into the expression
Now we substitute these values into the original expression:
\[ \cos 105^\circ + \sin 105^\circ = -\sin 15^\circ + \cos 15^\circ \]

### Step 4: Use the sine subtraction formula
We can rewrite \( -\sin 15^\circ + \cos 15^\circ \) as:
\[ \cos 15^\circ - \sin 15^\circ \]

### Step 5: Apply the sine subtraction formula
We can use the formula for the difference of sines:
\[ \cos A - \sin A = \sqrt{2} \sin\left(45^\circ - A\right) \]
where \( A = 15^\circ \). Thus:
\[ \cos 15^\circ - \sin 15^\circ = \sqrt{2} \sin(45^\circ - 15^\circ) = \sqrt{2} \sin(30^\circ) \]

### Step 6: Calculate the sine value
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ \cos 15^\circ - \sin 15^\circ = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2} \]

### Step 7: Final answer
Thus, the value of \( \cos 105^\circ + \sin 105^\circ \) is:
\[ \frac{\sqrt{2}}{2} \]

### Conclusion
The correct option is \( \frac{1}{\sqrt{2}} \), which corresponds to option 3.