The value of `(tan^(2)20^(@)-sin^(2)20^(@))/(tan^(2)20^(@).sin^(2)20^(@))` is

Options

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

Explanation

To solve the expression \((\tan^2 20^\circ - \sin^2 20^\circ) / (\tan^2 20^\circ \cdot \sin^2 20^\circ)\), we can follow these steps:

### Step 1: Rewrite \(\tan^2 20^\circ\)
We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore,
\[ \tan^2 20^\circ = \frac{\sin^2 20^\circ}{\cos^2 20^\circ} \]

### Step 2: Substitute \(\tan^2 20^\circ\) into the expression
Substituting this into our expression gives:
\[ \frac{\frac{\sin^2 20^\circ}{\cos^2 20^\circ} - \sin^2 20^\circ}{\frac{\sin^2 20^\circ}{\cos^2 20^\circ} \cdot \sin^2 20^\circ} \]

### Step 3: Simplify the numerator
The numerator can be rewritten as:
\[ \frac{\sin^2 20^\circ}{\cos^2 20^\circ} - \sin^2 20^\circ = \frac{\sin^2 20^\circ - \sin^2 20^\circ \cos^2 20^\circ}{\cos^2 20^\circ} \]
Taking the common denominator \(\cos^2 20^\circ\) in the numerator results in:
\[ \frac{\sin^2 20^\circ(1 - \cos^2 20^\circ)}{\cos^2 20^\circ} \]

### Step 4: Use the Pythagorean identity
Using the identity \(1 - \cos^2 \theta = \sin^2 \theta\), we can simplify further:
\[ \frac{\sin^2 20^\circ \cdot \sin^2 20^\circ}{\cos^2 20^\circ} \]

### Step 5: Rewrite the entire expression
Now, substituting this back into the full expression gives:
\[ \frac{\frac{\sin^4 20^\circ}{\cos^2 20^\circ}}{\frac{\sin^2 20^\circ}{\cos^2 20^\circ} \cdot \sin^2 20^\circ} \]

### Step 6: Simplify the denominator
The denominator simplifies to:
\[ \frac{\sin^4 20^\circ}{\cos^2 20^\circ} \]

### Step 7: Cancel out terms
Now we can cancel \(\sin^4 20^\circ\) in the numerator and denominator:
\[ \frac{\sin^4 20^\circ}{\sin^4 20^\circ} = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]