`cos7^(@)cos23^(@)cos45^(@)cosec83^(@)cosec67^(@)=?`

Options

A. 0
B. 1
C. `(1)/(2)`
D. `(1)/(sqrt(2))`

Answer

To solve the expression \( \cos 7^\circ \cos 23^\circ \cos 45^\circ \csc 83^\circ \csc 67^\circ \), we can use the complementary angle property of trigonometric functions. Here’s a step-by-step breakdown of the solution:

### Step 1: Rewrite Cosecant in Terms of Secant
Recall that \( \csc \theta = \sec(90^\circ - \theta) \). Therefore, we can rewrite:
\[ \csc 83^\circ = \sec(90^\circ - 83^\circ) = \sec 7^\circ \]
\[ \csc 67^\circ = \sec(90^\circ - 67^\circ) = \sec 23^\circ \]

### Step 2: Substitute into the Expression
Now, substitute these into the original expression:
\[ \cos 7^\circ \cos 23^\circ \cos 45^\circ \csc 83^\circ \csc 67^\circ = \cos 7^\circ \cos 23^\circ \cos 45^\circ \sec 7^\circ \sec 23^\circ \]

### Step 3: Use the Identity \( \cos \theta \sec \theta = 1 \)
Using the identity \( \cos \theta \sec \theta = 1 \):
\[ \cos 7^\circ \sec 7^\circ = 1 \]
\[ \cos 23^\circ \sec 23^\circ = 1 \]

### Step 4: Simplify the Expression
Now we can simplify the expression:
\[ 1 \cdot 1 \cdot \cos 45^\circ = \cos 45^\circ \]

### Step 5: Substitute the Value of \( \cos 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the final value of the expression is:
\[ \frac{1}{\sqrt{2}} \]