The value of `(2sin40^(@).sin50^(@).tan10^(@))/(cos 80^(@))` is

Options

A. `1//2`
B. 1
C. 2
D. none of these

Explanation

To solve the expression \(\frac{2 \sin 40^\circ \cdot \sin 50^\circ \cdot \tan 10^\circ}{\cos 80^\circ}\), we can follow these steps:

### Step 1: Rewrite \(\tan 10^\circ\)
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, we can rewrite \(\tan 10^\circ\) as:
\[ \tan 10^\circ = \frac{\sin 10^\circ}{\cos 10^\circ} \]

### Step 2: Substitute \(\tan 10^\circ\) into the expression
Substituting this into the expression gives:
\[ \frac{2 \sin 40^\circ \cdot \sin 50^\circ \cdot \frac{\sin 10^\circ}{\cos 10^\circ}}{\cos 80^\circ} \]

### Step 3: Simplify the expression
This can be simplified to:
\[ \frac{2 \sin 40^\circ \cdot \sin 50^\circ \cdot \sin 10^\circ}{\cos 10^\circ \cdot \cos 80^\circ} \]

### Step 4: Use the identity for \(\cos 80^\circ\)
We know that \(\cos 80^\circ = \sin 10^\circ\) (since \(\cos(90^\circ - \theta) = \sin \theta\)). Thus, we can replace \(\cos 80^\circ\) in the expression:
\[ \frac{2 \sin 40^\circ \cdot \sin 50^\circ \cdot \sin 10^\circ}{\cos 10^\circ \cdot \sin 10^\circ} \]

### Step 5: Cancel \(\sin 10^\circ\)
Now we can cancel \(\sin 10^\circ\) from the numerator and denominator:
\[ \frac{2 \sin 40^\circ \cdot \sin 50^\circ}{\cos 10^\circ} \]

### Step 6: Use the product-to-sum identities
We can use the product-to-sum identities, specifically:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
Let \(A = 40^\circ\) and \(B = 50^\circ\):
\[ 2 \sin 40^\circ \sin 50^\circ = \cos(40^\circ - 50^\circ) - \cos(40^\circ + 50^\circ) = \cos(-10^\circ) - \cos(90^\circ) \]
Since \(\cos(-10^\circ) = \cos(10^\circ)\) and \(\cos(90^\circ) = 0\), we have:
\[ 2 \sin 40^\circ \sin 50^\circ = \cos(10^\circ) \]

### Step 7: Substitute back into the expression
Now substituting back into our expression gives:
\[ \frac{\cos(10^\circ)}{\cos(10^\circ)} \]

### Step 8: Final simplification
This simplifies to:
\[ 1 \]

Thus, the value of the expression \(\frac{2 \sin 40^\circ \cdot \sin 50^\circ \cdot \tan 10^\circ}{\cos 80^\circ}\) is \(1\).

### Final Answer
The value is \(1\).

---