If A + B = C, then tan A tan B tan C = ?
tan C - tan A - tan B
Understanding the Trigonometric Identity Problem
The question asks us to find the value of the product \( \tan A \tan B \tan C \) given the condition that the sum of two angles, \( A \) and \( B \), is equal to a third angle, \( C \). This means we are given \( A + B = C \). We need to use trigonometric identities to relate the tangents of these angles.
Deriving the Relationship between Tangents
We start with the given condition:
\[ A + B = C \]To involve the tangent function, we can take the tangent of both sides of this equation:
\[ \tan(A + B) = \tan C \]Now, we use the tangent addition formula, which states that \( \tan(X + Y) = \frac{\tan X + \tan Y}{1 - \tan X \tan Y} \). Applying this formula to the left side of our equation with \( X = A \) and \( Y = B \), we get:
\[ \frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan C \]Assuming \( 1 - \tan A \tan B \neq 0 \), we can multiply both sides by \( (1 - \tan A \tan B) \):
\[ \tan A + \tan B = \tan C (1 - \tan A \tan B) \]Next, we distribute \( \tan C \) on the right side:
\[ \tan A + \tan B = \tan C - \tan A \tan B \tan C \]Our goal is to find the expression for \( \tan A \tan B \tan C \). We can rearrange the equation to isolate this term. Let's move \( - \tan A \tan B \tan C \) to the left side and \( \tan A + \tan B \) to the right side:
\[ \tan A \tan B \tan C = \tan C - (\tan A + \tan B) \] \[ \tan A \tan B \tan C = \tan C - \tan A - \tan B \]This gives us the relationship between \( \tan A \tan B \tan C \) and \( \tan A, \tan B, \tan C \).
Analyzing the Options
We found that if \( A + B = C \), then \( \tan A \tan B \tan C = \tan C - \tan A - \tan B \). Let's compare this result with the given options:
- Option 1: \( \tan C + \tan A - \tan B \) - This does not match our result.
- Option 2: \( \tan C + \tan A + \tan B \) - This does not match our result.
- Option 3: \( \tan A - \tan B - \tan C \) - This does not match our result.
- Option 4: \( \tan C - \tan A - \tan B \) - This exactly matches our derived result.
Therefore, the correct expression for \( \tan A \tan B \tan C \) is \( \tan C - \tan A - \tan B \).
Step-by-Step Solution
- Start with the given condition \( A + B = C \).
- Take the tangent of both sides: \( \tan(A + B) = \tan C \).
- Apply the tangent addition formula \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) to get \( \frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan C \).
- Multiply both sides by \( (1 - \tan A \tan B) \): \( \tan A + \tan B = \tan C (1 - \tan A \tan B) \).
- Distribute \( \tan C \) on the right side: \( \tan A + \tan B = \tan C - \tan A \tan B \tan C \).
- Rearrange the equation to isolate \( \tan A \tan B \tan C \): \( \tan A \tan B \tan C = \tan C - \tan A - \tan B \).
| Step | Equation | Identity/Operation Used |
|---|---|---|
| 1 | \( A + B = C \) | Given Condition |
| 2 | \( \tan(A + B) = \tan C \) | Taking tangent of both sides |
| 3 | \( \frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan C \) | Tangent Addition Formula |
| 4 | \( \tan A + \tan B = \tan C (1 - \tan A \tan B) \) | Multiply by denominator |
| 5 | \( \tan A + \tan B = \tan C - \tan A \tan B \tan C \) | Distribute \( \tan C \) |
| 6 | \( \tan A \tan B \tan C = \tan C - \tan A - \tan B \) | Rearrangement |
Revision Table: Key Trigonometric Identities
| Identity | Formula |
|---|---|
| Tangent Addition | \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \) |
| Tangent Subtraction | \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \) |
| Sine Addition | \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) |
| Cosine Addition | \( \cos(A + B) = \cos A \cos B - \sin A \sin B \) |
Additional Information on Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every single value of the occurring variables for which both sides of the equation are defined. They are fundamental in simplifying expressions, solving trigonometric equations, and proving other identities. The identity \( \tan A \tan B \tan C = \tan C - \tan A - \tan B \) when \( A + B = C \) is a useful result that often appears in problems involving triangle properties (since the sum of angles in a triangle is \( \pi \) or \( 180^\circ \)) or general angle relationships.
It's important to remember the conditions under which these identities are valid. For the tangent function, angles should not be odd multiples of \( \pi/2 \) (\( 90^\circ \)), where the tangent is undefined. In the derivation above, we also assumed that \( 1 - \tan A \tan B \neq 0 \), which means \( \tan A \tan B \neq 1 \). If \( \tan A \tan B = 1 \), then \( \tan(A+B) \) is undefined, which would imply \( A+B \) is an odd multiple of \( \pi/2 \). If \( C \) is also an odd multiple of \( \pi/2 \), \( \tan C \) is undefined. So, the identity holds when all tangent values are defined.
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