Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\frac{\tan (A+B)-\tan B}{1+\tan (A+B) \tan B}=\tan A$$

Step-by-Step Solution

Answer

The given equation is an identity: the equation reduces to \( \tan A \).

Step 1: Recall tangent sum formula

First, we need to recall the tangent sum formula. The formula for \( \tan(A+B) \) is: \[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]. We will use this formula to simplify \( \tan(A+B) \) in the given expression.

Step 2: Substitute the tangent sum formula into the equation

Substitute \( \tan(A+B) \) with \( \frac{\tan A + \tan B}{1 - \tan A \tan B} \) in the equation: \[ \frac{\frac{\tan A + \tan B}{1 - \tan A \tan B} - \tan B}{1 + \frac{\tan A + \tan B}{1 - \tan A \tan B} \cdot \tan B} = \tan A \].

Step 3: Simplify the numerator of the fraction

Now focus on the numerator of the expression: \[ \frac{\tan A + \tan B}{1 - \tan A \tan B} - \tan B \]. Simplify it to get: \[ \frac{\tan A + \tan B - \tan B (1 - \tan A \tan B)}{1 - \tan A \tan B} = \frac{\tan A + \tan B - \tan B + \tan B \tan A \tan B}{1 - \tan A \tan B} \]. The simplification gives us: \[ \frac{\tan A (1+\tan^2 B)}{1 - \tan A \tan B} \].

Step 4: Simplify the denominator of the fraction

Next, simplify the denominator: \[ 1 + \frac{\tan A + \tan B}{1 - \tan A \tan B} \tan B \]. This becomes \[ \frac{1 - \tan A \tan B + (\tan A + \tan B) \tan B}{1 - \tan A \tan B} \], which simplifies to \[ \frac{1 - \tan A \tan B + \tan A \tan B + \tan^2 B}{1 - \tan A \tan B} = \frac{1 + \tan^2 B}{1 - \tan A \tan B} \].

Step 5: Verify the identity

Now plug the simplified numerator and denominator back into the main fraction: \[ \frac{\frac{\tan A (1+\tan^2 B)}{1 - \tan A \tan B}}{\frac{1 + \tan^2 B}{1 - \tan A \tan B}} = \tan A \]. On simplifying, this equates to \[ \frac{\tan A (1+\tan^2 B)}{1 + \tan^2 B} = \tan A \]. Indeed, this simplifies to \( \tan A \), verifying that the original expression is an identity.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. When dealing with right triangles, the tangent of an angle represents the ratio of the opposite side to the adjacent side:

• Opposite Side: The side opposite to the angle in question.
• Adjacent Side: The side next to the angle, forming a right-angled triangle, excluding the hypotenuse.

Mathematically, it is expressed as: \[tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}.\]
Furthermore, in the unit circle approach, tangent can be understood as the y-coordinate divided by the x-coordinate of a point on the unit circle corresponding to an angle \( \theta \). This relationship opens up deeper trigonometric identities and properties useful in solving equations and verifying identities in trigonometry.
The tangent function plays a crucial role when investigating properties such as periodicity and symmetry, and it is especially important in calculus when considering derivatives and integrals of trigonometric functions.

Angle Sum Formulas

The angle sum formulas are crucial for adding or subtracting angles in trigonometry. Specifically, the formula for the tangent of a sum of two angles \( A \) and \( B \) is:
\[\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}.\]This formula is particularly useful when you need to express the tangent of a sum in terms of individual tangents.
These sum formulas reveal that the tangent of an angle is not simply the sum of the tangents but involves a complex interaction of multiple angles.
When verifying identities or solving equations involving

• angles of 90 degrees apart
• complicated sum differences

this set of formulas helps transform and simplify expressions. Understanding these relationships is pivotal in both algebraic manipulation and geometric interpretations of trigonometry.

Identity Verification

Identity verification in trigonometry involves proving that two trigonometric expressions are equivalent under all permissible conditions. This requires strategic manipulations using known identities and algebraic rules.
For example, in the given exercise, our goal is to prove the equation

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

is true for any angle \(A\) and \(B\). This begins with applying known trigonometric identities such as the tangent sum formula to transform the expressions.
Throughout the verification process, it's vital to identify and match similar terms through cancellation or strategic factoring. By the end, the left side must simplify entirely to the right side, confirming the expression as an identity. Identity verification builds foundational skills in formal mathematical proofs and ensures comprehension of trigonometric functions beyond simple computations.