`(cos 20^(@)+8 sin 70^(@) sin 50^(@) sin 10^(@))/( sin^(2)80^(@))` is equal to

Options

A. 1
B. 2
C. `3//4`
D. None of these

Answer

To solve the expression \((\cos 20^\circ + 8 \sin 70^\circ \sin 50^\circ \sin 10^\circ) / (\sin^2 80^\circ)\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ \frac{\cos 20^\circ + 8 \sin 70^\circ \sin 50^\circ \sin 10^\circ}{\sin^2 80^\circ} \]

2. **Use the identity for \(\sin 70^\circ\)**:
Since \(\sin 70^\circ = \cos 20^\circ\), we can substitute:
\[ \sin 70^\circ = \cos 20^\circ \]
Thus, the expression becomes:
\[ \frac{\cos 20^\circ + 8 \cos 20^\circ \sin 50^\circ \sin 10^\circ}{\sin^2 80^\circ} \]

3. **Factor out \(\cos 20^\circ\)**:
\[ = \frac{\cos 20^\circ (1 + 8 \sin 50^\circ \sin 10^\circ)}{\sin^2 80^\circ} \]

4. **Use the identity for \(\sin 80^\circ\)**:
We know that \(\sin 80^\circ = \cos 10^\circ\), so:
\[ \sin^2 80^\circ = \cos^2 10^\circ \]
The expression now looks like:
\[ = \frac{\cos 20^\circ (1 + 8 \sin 50^\circ \sin 10^\circ)}{\cos^2 10^\circ} \]

5. **Use the product-to-sum identities**:
We can use the identity:
\[ \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \]
For \(\sin 50^\circ \sin 10^\circ\):
\[ \sin 50^\circ \sin 10^\circ = \frac{1}{2} [\cos(50^\circ - 10^\circ) - \cos(50^\circ + 10^\circ)] \]
This simplifies to:
\[ = \frac{1}{2} [\cos 40^\circ - \cos 60^\circ] \]

6. **Substituting back**:
Substitute this back into the expression:
\[ = \frac{\cos 20^\circ \left(1 + 8 \cdot \frac{1}{2} [\cos 40^\circ - \frac{1}{2}]\right)}{\cos^2 10^\circ} \]
Simplifying further:
\[ = \frac{\cos 20^\circ \left(1 + 4 (\cos 40^\circ - \frac{1}{2})\right)}{\cos^2 10^\circ} \]
\[ = \frac{\cos 20^\circ \left(1 + 4 \cos 40^\circ - 2\right)}{\cos^2 10^\circ} \]
\[ = \frac{\cos 20^\circ (4 \cos 40^\circ - 1)}{\cos^2 10^\circ} \]

7. **Final simplification**:
Using the identity \(\cos 40^\circ = 2 \cos^2 20^\circ - 1\):
\[ = \frac{\cos 20^\circ (4(2 \cos^2 20^\circ - 1) - 1)}{\cos^2 10^\circ} \]
\[ = \frac{\cos 20^\circ (8 \cos^2 20^\circ - 4 - 1)}{\cos^2 10^\circ} \]
\[ = \frac{\cos 20^\circ (8 \cos^2 20^\circ - 5)}{\cos^2 10^\circ} \]

8. **Evaluate the final expression**:
After simplifying, we find that the entire expression evaluates to \(2\).

### Final Result:
\[ \frac{\cos 20^\circ + 8 \sin 70^\circ \sin 50^\circ \sin 10^\circ}{\sin^2 80^\circ} = 2 \]