What is `tan25^@tan15^@ + tan15^@ tan 50^@ + tan25^@ tan50^@` equal to ?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the expression \( \tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ \]

### Step 2: Group terms
We can group the terms involving \( \tan 15^\circ \) and \( \tan 50^\circ \):
\[ = \tan 15^\circ (\tan 25^\circ + \tan 50^\circ) + \tan 25^\circ \tan 50^\circ \]

### Step 3: Use the identity for tangent
We know that:
\[ \tan(90^\circ - x) = \cot x \]
Thus, \( \tan 50^\circ = \cot 40^\circ \).

### Step 4: Substitute and simplify
Now, we can rewrite \( \tan 50^\circ \) as \( \cot 40^\circ \):
\[ = \tan 15^\circ (\tan 25^\circ + \cot 40^\circ) + \tan 25^\circ \cot 40^\circ \]

### Step 5: Use the identity for \( \tan a + \cot a \)
We can use the identity:
\[ \tan a + \cot a = \frac{\sin^2 a + \cos^2 a}{\sin a \cos a} = \frac{1}{\sin a \cos a} \]
Thus, we can express \( \tan 25^\circ + \cot 40^\circ \) in terms of sine and cosine.

### Step 6: Combine terms
Now we can combine the terms:
\[ \tan 15^\circ \tan 25^\circ + \tan 15^\circ \cot 40^\circ + \tan 25^\circ \cot 40^\circ \]

### Step 7: Apply the tangent addition formula
Using the tangent addition formula:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
We can apply it to \( \tan(15^\circ + 25^\circ) \) and simplify.

### Step 8: Final simplification
After simplification, we find that:
\[ \tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ = 1 \]

### Final Answer
Thus, the final result is:
\[ \boxed{1} \]