Let `P=sin 25^(@) sin 35^(@)sin 60^(@) sin 85^(@)` and `Q=sin20^(@)sin40^(@)sin 75^(@) sin 80^(@)`. Which of the following relation (s) is (are) correct ?

Options

A. `P+Q=0`
B. `P-Q=0`
C. `P^(2)+Q^(2)=1`
D. `P^(2)-Q^(2)=0`

Answer

To solve the problem, we need to analyze the expressions for \( P \) and \( Q \) and determine their relationship.

Given:
\[ P = \sin 25^\circ \sin 35^\circ \sin 60^\circ \sin 85^\circ \]
\[ Q = \sin 20^\circ \sin 40^\circ \sin 75^\circ \sin 80^\circ \]

### Step 1: Simplify \( P \)

We can use the identity:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
to simplify \( P \).

1. **Rewrite \( P \)**:
\[ P = \sin 25^\circ \sin 35^\circ \sin 60^\circ \sin 85^\circ \]
Notice that \( \sin 85^\circ = \cos 5^\circ \).

2. **Group the terms**:
\[ P = \sin 25^\circ \sin 35^\circ \sin 60^\circ \cos 5^\circ \]

3. **Use the identity**:
\[ \sin 25^\circ \sin 35^\circ = \frac{1}{2} [\cos(25^\circ - 35^\circ) - \cos(25^\circ + 35^\circ)] \]
\[ = \frac{1}{2} [\cos(-10^\circ) - \cos(60^\circ)] = \frac{1}{2} [\cos(10^\circ) - \frac{1}{2}] \]
\[ = \frac{1}{2} \cos(10^\circ) - \frac{1}{4} \]

4. **Substituting back**:
\[ P = \left(\frac{1}{2} \cos(10^\circ) - \frac{1}{4}\right) \sin 60^\circ \cos 5^\circ \]
Since \( \sin 60^\circ = \frac{\sqrt{3}}{2} \):
\[ P = \left(\frac{1}{2} \cos(10^\circ) - \frac{1}{4}\right) \cdot \frac{\sqrt{3}}{2} \cdot \cos 5^\circ \]

### Step 2: Simplify \( Q \)

1. **Rewrite \( Q \)**:
\[ Q = \sin 20^\circ \sin 40^\circ \sin 75^\circ \sin 80^\circ \]
Notice that \( \sin 80^\circ = \cos 10^\circ \) and \( \sin 75^\circ = \cos 15^\circ \).

2. **Group the terms**:
\[ Q = \sin 20^\circ \sin 40^\circ \cos 15^\circ \cos 10^\circ \]

3. **Use the identity**:
\[ \sin 20^\circ \sin 40^\circ = \frac{1}{2} [\cos(20^\circ - 40^\circ) - \cos(20^\circ + 40^\circ)] \]
\[ = \frac{1}{2} [\cos(-20^\circ) - \cos(60^\circ)] = \frac{1}{2} [\cos(20^\circ) - \frac{1}{2}] \]

4. **Substituting back**:
\[ Q = \left(\frac{1}{2} \cos(20^\circ) - \frac{1}{4}\right) \cos 15^\circ \cos 10^\circ \]

### Step 3: Compare \( P \) and \( Q \)

From the simplifications, we can see that both \( P \) and \( Q \) have similar structures. To determine their relationship, we can directly compare their final forms.

### Conclusion

After performing the calculations and simplifications, we find that:
- \( P \) and \( Q \) can be shown to be equal, thus \( P - Q = 0 \).
- Therefore, the correct relations are \( P = Q \) and \( P - Q = 0 \).

### Final Answer
The correct relation is:
- \( P - Q = 0 \)