What is the value of cos `1485^@ xx "cosec" 855^@` ?

Answer

To find the value of \( \cos 1485^\circ \times \csc 855^\circ \), we will simplify each term step by step.

### Step 1: Simplifying \( \cos 1485^\circ \)

To simplify \( \cos 1485^\circ \), we can use the periodic property of cosine, which states that \( \cos(\theta) = \cos(\theta + 360n) \) for any integer \( n \).

First, we can reduce \( 1485^\circ \) by subtracting \( 360^\circ \) until we get an angle within the range \( 0^\circ \) to \( 360^\circ \).

\[ 1485^\circ - 4 \times 360^\circ = 1485^\circ - 1440^\circ = 45^\circ \]

Thus,

\[ \cos 1485^\circ = \cos 45^\circ \]

### Step 2: Finding \( \cos 45^\circ \)

From trigonometric values, we know:

\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 3: Simplifying \( \csc 855^\circ \)

Next, we simplify \( \csc 855^\circ \). The cosecant function is the reciprocal of the sine function, so we can write:

\[ \csc 855^\circ = \frac{1}{\sin 855^\circ} \]

Now, we can reduce \( 855^\circ \) similarly by subtracting \( 360^\circ \):

\[ 855^\circ - 2 \times 360^\circ = 855^\circ - 720^\circ = 135^\circ \]

Thus,

\[ \csc 855^\circ = \csc 135^\circ = \frac{1}{\sin 135^\circ} \]

### Step 4: Finding \( \sin 135^\circ \)

Using the property of sine, we know:

\[ \sin 135^\circ = \sin(180^\circ - 45^\circ) = \sin 45^\circ = \frac{1}{\sqrt{2}} \]

Therefore,

\[ \csc 135^\circ = \frac{1}{\sin 135^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 5: Combining the Results

Now we can combine the results from Steps 2 and 4:

\[ \cos 1485^\circ \times \csc 855^\circ = \frac{1}{\sqrt{2}} \times \sqrt{2} \]

This simplifies to:

\[ \frac{1}{\sqrt{2}} \times \sqrt{2} = 1 \]

### Final Answer

Thus, the value of \( \cos 1485^\circ \times \csc 855^\circ \) is:

\[ \boxed{1} \]