`tan 20^(@) + tan 25^(@) +tan 20^@ tan25^(@)=`

Options

A. `1/sqrt2`
B. `-1/sqrt2`
C. 1
D. `sqrt2`

Answer

To solve the expression \( \tan 20^\circ + \tan 25^\circ + \tan 20^\circ \tan 25^\circ \), we can use the formula for the tangent of the sum of two angles.

### Step 1: Recall the formula for tangent of sum
The formula for the tangent of the sum of two angles \( A \) and \( B \) is given by:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
In our case, we can let \( A = 20^\circ \) and \( B = 25^\circ \).

### Step 2: Apply the formula
We can express \( \tan(20^\circ + 25^\circ) \):
\[ \tan(45^\circ) = \tan(20^\circ + 25^\circ) \]
Since \( 20^\circ + 25^\circ = 45^\circ \), we know that:
\[ \tan(45^\circ) = 1 \]

### Step 3: Substitute into the formula
Using the formula, we have:
\[ 1 = \frac{\tan 20^\circ + \tan 25^\circ}{1 - \tan 20^\circ \tan 25^\circ} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ 1 - \tan 20^\circ \tan 25^\circ = \tan 20^\circ + \tan 25^\circ \]

### Step 5: Rearranging the equation
Rearranging the equation, we get:
\[ \tan 20^\circ + \tan 25^\circ + \tan 20^\circ \tan 25^\circ = 1 \]

### Conclusion
Thus, the value of the expression \( \tan 20^\circ + \tan 25^\circ + \tan 20^\circ \tan 25^\circ \) is:
\[ \boxed{1} \]