`(tan8 0^(@)-tan1 0^(@))/tan70^(@) `

Options

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

Answer

To solve the expression \((\tan 80^\circ - \tan 10^\circ) / \tan 70^\circ\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\tan 80^\circ - \tan 10^\circ}{\tan 70^\circ} \]

### Step 2: Use the identity for tangent
Recall that \(\tan(90^\circ - x) = \cot x\). Thus, we can rewrite \(\tan 80^\circ\) as:
\[ \tan 80^\circ = \cot 10^\circ \]
So, we can substitute this into our expression:
\[ \frac{\cot 10^\circ - \tan 10^\circ}{\tan 70^\circ} \]

### Step 3: Rewrite \(\tan 70^\circ\)
Using the same identity, we can also express \(\tan 70^\circ\) as:
\[ \tan 70^\circ = \cot 20^\circ \]
Thus, the expression becomes:
\[ \frac{\cot 10^\circ - \tan 10^\circ}{\cot 20^\circ} \]

### Step 4: Convert cotangent to tangent
We know that \(\cot x = \frac{1}{\tan x}\). Therefore:
\[ \cot 10^\circ = \frac{1}{\tan 10^\circ} \]
Substituting this into the expression gives:
\[ \frac{\frac{1}{\tan 10^\circ} - \tan 10^\circ}{\cot 20^\circ} \]

### Step 5: Simplify the numerator
To simplify the numerator \(\frac{1}{\tan 10^\circ} - \tan 10^\circ\), we can find a common denominator:
\[ \frac{1 - \tan^2 10^\circ}{\tan 10^\circ} \]
Thus, the expression now becomes:
\[ \frac{\frac{1 - \tan^2 10^\circ}{\tan 10^\circ}}{\cot 20^\circ} \]

### Step 6: Substitute \(\cot 20^\circ\)
Since \(\cot 20^\circ = \frac{1}{\tan 20^\circ}\), we rewrite the expression:
\[ \frac{1 - \tan^2 10^\circ}{\tan 10^\circ} \cdot \tan 20^\circ \]

### Step 7: Use the double angle formula
Using the double angle formula for tangent, we know:
\[ \tan 20^\circ = \frac{2 \tan 10^\circ}{1 - \tan^2 10^\circ} \]
Substituting this back into our expression gives:
\[ \frac{1 - \tan^2 10^\circ}{\tan 10^\circ} \cdot \frac{2 \tan 10^\circ}{1 - \tan^2 10^\circ} \]

### Step 8: Simplify the expression
The \((1 - \tan^2 10^\circ)\) terms cancel out:
\[ 2 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{2} \]