What is sin `105^@ + cos105^@` equal to:

Options

A. `sin 150^@`
B. `cos50^@`
C. `1//sqrt2`
D. 0

Answer

To solve the problem of finding \( \sin 105^\circ + \cos 105^\circ \), we can use the angle addition formulas for sine and cosine.

### Step-by-step Solution:

1. **Rewrite the Angles**:
We can express \( 105^\circ \) as \( 60^\circ + 45^\circ \):
\[ \sin 105^\circ = \sin(60^\circ + 45^\circ) \]
\[ \cos 105^\circ = \cos(60^\circ + 45^\circ) \]

2. **Apply the Sine Addition Formula**:
Using the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \):
\[ \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \]

3. **Substitute Known Values**:
We know:
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)

Substituting these values:
\[ \sin 105^\circ = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{2}} \]
\[ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} \]

4. **Combine the Terms**:
Combine the fractions:
\[ \sin 105^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

5. **Apply the Cosine Addition Formula**:
Using the cosine addition formula \( \cos(a + b) = \cos a \cos b - \sin a \sin b \):
\[ \cos(60^\circ + 45^\circ) = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ \]
Substituting the known values:
\[ \cos 105^\circ = \frac{1}{2} \cdot \frac{1}{\sqrt{2}} - \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} \]
\[ = \frac{1 - \sqrt{3}}{2\sqrt{2}} \]

6. **Combine Sine and Cosine**:
Now we can combine \( \sin 105^\circ + \cos 105^\circ \):
\[ \sin 105^\circ + \cos 105^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} + \frac{1 - \sqrt{3}}{2\sqrt{2}} \]
\[ = \frac{(\sqrt{3} + 1) + (1 - \sqrt{3})}{2\sqrt{2}} \]
\[ = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Final Answer:
Thus, \( \sin 105^\circ + \cos 105^\circ = \frac{1}{\sqrt{2}} \).