The value of `(tan7 0^0-tan2 0^0)/(tan5 0^0)=`

Options

A. 2
B. 1
C. 0
D. 3

Answer

To solve the expression \((\tan 70^\circ - \tan 20^\circ) / \tan 50^\circ\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\tan 70^\circ - \tan 20^\circ}{\tan 50^\circ} \]
We can rewrite \(\tan 50^\circ\) as \(\tan(70^\circ - 20^\circ)\).

### Step 2: Apply the tangent subtraction formula
Using the tangent subtraction formula, \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\), we can express \(\tan 50^\circ\):
\[ \tan 50^\circ = \tan(70^\circ - 20^\circ) = \frac{\tan 70^\circ - \tan 20^\circ}{1 + \tan 70^\circ \tan 20^\circ} \]

### Step 3: Substitute back into the expression
Now we substitute this back into our original expression:
\[ \frac{\tan 70^\circ - \tan 20^\circ}{\tan 50^\circ} = \frac{\tan 70^\circ - \tan 20^\circ}{\frac{\tan 70^\circ - \tan 20^\circ}{1 + \tan 70^\circ \tan 20^\circ}} \]

### Step 4: Simplify the expression
When we simplify this, we can cancel \((\tan 70^\circ - \tan 20^\circ)\) in the numerator and denominator:
\[ = 1 + \tan 70^\circ \tan 20^\circ \]

### Step 5: Use the complementary angle identity
Since \(\tan(90^\circ - \theta) = \cot \theta\), we have:
\[ \tan 70^\circ = \cot 20^\circ \]
Thus, we can rewrite:
\[ \tan 70^\circ \tan 20^\circ = \cot 20^\circ \tan 20^\circ = 1 \]

### Step 6: Final result
Substituting back, we get:
\[ 1 + \tan 70^\circ \tan 20^\circ = 1 + 1 = 2 \]

Therefore, the value of \(\frac{\tan 70^\circ - \tan 20^\circ}{\tan 50^\circ}\) is:
\[ \boxed{2} \]