The value of `tan65^(@) - 2tan40^(@)` is equal to -

Options

A. `-tan 20^(@)`
B. `tan 20^(@)`
C. `tan 25^(@)`
D. `tan 40^(@)`

Explanation

To find the value of \( \tan 65^\circ - 2 \tan 40^\circ \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \tan 65^\circ - 2 \tan 40^\circ \]

### Step 2: Rewrite \( 2 \tan 40^\circ \)
We can express \( 2 \tan 40^\circ \) as \( \tan 40^\circ + \tan 40^\circ \):
\[ \tan 65^\circ - \tan 40^\circ - \tan 40^\circ \]

### Step 3: Apply the tangent subtraction formula
We can use the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Let \( A = 65^\circ \) and \( B = 40^\circ \). Thus, we can rewrite the expression as:
\[ \tan(65^\circ - 40^\circ) = \tan 25^\circ \]
So we have:
\[ \tan 65^\circ - \tan 40^\circ = \tan 25^\circ (1 + \tan 65^\circ \tan 40^\circ) \]

### Step 4: Substitute back into the expression
Now, substituting back into our expression:
\[ \tan 65^\circ - 2 \tan 40^\circ = \tan 25^\circ (1 + \tan 65^\circ \tan 40^\circ) - \tan 40^\circ \]

### Step 5: Simplify the expression
Since \( \tan 40^\circ \) appears in both terms, we can simplify:
\[ \tan 25^\circ + \tan 65^\circ \tan 40^\circ - \tan 40^\circ \]
This leads us to:
\[ \tan 25^\circ \]

### Conclusion
Thus, the value of \( \tan 65^\circ - 2 \tan 40^\circ \) is:
\[ \tan 25^\circ \]