Understanding the Trigonometry Problem

The question asks for the value of a specific trigonometric expression involving the tangent of three angles: \(25^\circ\), \(15^\circ\), and \(50^\circ\). The expression is \( \tan(25^\circ)\tan(15^\circ) + \tan(15^\circ)\tan(50^\circ) + \tan(25^\circ)\tan(50^\circ) \). To solve this, we need to look for relationships between these angles that can help simplify the expression using trigonometric identities.

Analyzing the Angles

Let's examine the given angles: \( A = 25^\circ \), \( B = 15^\circ \), and \( C = 50^\circ \). Let's find their sum:

\( A + B + C = 25^\circ + 15^\circ + 50^\circ = 90^\circ \)

The sum of the three angles is \(90^\circ\). This is a crucial observation, as there is a well-known trigonometric identity related to the tangents of angles whose sum is \(90^\circ\).

Using the Angle Sum Identity for Tangent

Consider three angles \(A\), \(B\), and \(C\) such that \( A + B + C = 90^\circ \). We can write \( A + B = 90^\circ - C \).

Taking the tangent of both sides of the equation \( A + B = 90^\circ - C \):

\( \tan(A + B) = \tan(90^\circ - C) \)

We know the tangent addition formula: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

We also know the complementary angle identity: \( \tan(90^\circ - C) = \cot C = \frac{1}{\tan C} \) (assuming \( \tan C \neq 0 \)).

Equating the two expressions:

\( \frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{1}{\tan C} \)

Now, we cross-multiply:

\( \tan C (\tan A + \tan B) = 1 \cdot (1 - \tan A \tan B) \)

\( \tan A \tan C + \tan B \tan C = 1 - \tan A \tan B \)

Rearranging the terms to group all tangent product terms on one side:

\( \tan A \tan B + \tan A \tan C + \tan B \tan C = 1 \)

This identity states that if \( A + B + C = 90^\circ \), then \( \tan A \tan B + \tan B \tan C + \tan A \tan C = 1 \).

Applying the Identity to the Given Expression

In our problem, the angles are \( A = 25^\circ \), \( B = 15^\circ \), and \( C = 50^\circ \). We confirmed that \( A + B + C = 25^\circ + 15^\circ + 50^\circ = 90^\circ \).

The given expression is \( \tan(25^\circ)\tan(15^\circ) + \tan(15^\circ)\tan(50^\circ) + \tan(25^\circ)\tan(50^\circ) \).

Comparing this with the identity \( \tan A \tan B + \tan B \tan C + \tan A \tan C \), we see that it matches exactly with \( A = 25^\circ \), \( B = 15^\circ \), and \( C = 50^\circ \).

Since the condition \( A + B + C = 90^\circ \) is satisfied, the value of the expression must be equal to 1 according to the identity.

Calculating the Value

Using the identity \( \tan A \tan B + \tan B \tan C + \tan A \tan C = 1 \) for \( A+B+C=90^\circ \):

\( \tan(25^\circ)\tan(15^\circ) + \tan(15^\circ)\tan(50^\circ) + \tan(25^\circ)\tan(50^\circ) = 1 \)

Conclusion

The value of the expression \( \tan25^\circ\tan15^\circ + tan15^\circ tan50^\circ + tan25^\circ tan50^\circ \) is 1.

Revision Table: Trigonometric Identities

Additional Information: Related Tangent Identities

The identity used in this problem, \( \tan A \tan B + \tan B \tan C + \tan A \tan C = 1 \) when \( A+B+C=90^\circ \), is a useful result derived from the tangent addition formula and complementary angle identities.

Another related identity involves cotangents when the sum of angles is \(180^\circ\). If \( A+B+C = 180^\circ \), then \( \tan A + \tan B + \tan C = \tan A \tan B \tan C \).

Understanding these identities helps in simplifying complex trigonometric expressions and solving problems more efficiently.