The value of `cos105^(@)+sin105^(@)` is

Options

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

Answer

To find the value of \( \cos 105^\circ + \sin 105^\circ \), we can break it down into two parts: calculating \( \cos 105^\circ \) and \( \sin 105^\circ \) separately, and then adding the results together.

### Step 1: Calculate \( \cos 105^\circ \)

We can express \( 105^\circ \) as \( 60^\circ + 45^\circ \).

Using the cosine addition formula:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
where \( a = 60^\circ \) and \( b = 45^\circ \).

Substituting the values:
\[ \cos 105^\circ = \cos(60^\circ + 45^\circ) = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ \]

### Step 2: Substitute known values

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

Substituting these values into the equation:
\[ \cos 105^\circ = \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) \]

### Step 3: Simplify the expression

Calculating the first term:
\[ \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]

Calculating the second term:
\[ \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{2\sqrt{2}} \]

Putting it all together:
\[ \cos 105^\circ = \frac{1}{2\sqrt{2}} - \frac{\sqrt{3}}{2\sqrt{2}} = \frac{1 - \sqrt{3}}{2\sqrt{2}} \]

### Step 4: Calculate \( \sin 105^\circ \)

Now, we can express \( 105^\circ \) as \( 60^\circ + 45^\circ \) again but using the sine addition formula:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]

Substituting the values:
\[ \sin 105^\circ = \sin(60^\circ + 45^\circ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \]

### Step 5: Substitute known values again

Using the same values as before:
\[ \sin 105^\circ = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) \]

### Step 6: Simplify the expression

Calculating the first term:
\[ \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{2\sqrt{2}} \]

Calculating the second term:
\[ \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]

Putting it all together:
\[ \sin 105^\circ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 7: Add \( \cos 105^\circ \) and \( \sin 105^\circ \)

Now we add the two results:
\[ \cos 105^\circ + \sin 105^\circ = \frac{1 - \sqrt{3}}{2\sqrt{2}} + \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 8: Combine the fractions

Since the denominators are the same, we can combine the numerators:
\[ \cos 105^\circ + \sin 105^\circ = \frac{(1 - \sqrt{3}) + (\sqrt{3} + 1)}{2\sqrt{2}} = \frac{1 - \sqrt{3} + \sqrt{3} + 1}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the value of \( \cos 105^\circ + \sin 105^\circ \) is:
\[ \frac{1}{\sqrt{2}} \]