Problem 56 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The given equation is verified as an identity.

Step 1: Recall the Formula for Sine of Sum

The identity for \(\sin(A + B)\) is \(\sin(A)\cos(B) + \cos(A)\sin(B)\). We will use this to start verifying the given equation.

Step 2: Substitute the Identity into the Equation

Substitute \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\) into the left side of the equation: \(\frac{\sin(A)\cos(B) + \cos(A)\sin(B)}{\cos A \cos B}\).

Step 3: Simplify Using Basic Algebra

Split the fraction: \(\frac{\sin(A)\cos(B)}{\cos A \cos B} + \frac{\cos(A)\sin(B)}{\cos A \cos B}\).

Step 4: Simplify Each Fraction

Simplify each individual fraction: \(\frac{\sin(A)\cos(B)}{\cos A \cos B} = \frac{\sin(A)}{\cos(A)}\) and \(\frac{\cos(A)\sin(B)}{\cos A \cos B} = \frac{\sin(B)}{\cos(B)}\).

Step 5: Recognize Trigonometric Functions

Recognize that \(\frac{\sin(A)}{\cos(A)} = \tan(A)\) and \(\frac{\sin(B)}{\cos(B)} = \tan(B)\).

Step 6: Verify the Identity

Combine the results: \(\tan(A) + \tan(B)\). This matches the right side of the original equation, confirming the identity as \(\frac{\sin(A+B)}{\cos A \cos B} = \tan A + \tan B\).

Key Concepts

Sine and Cosine Formulas

The Sine and Cosine formulas are foundational concepts in trigonometry, allowing us to express complex relationships between different angles. In the equation we are examining, the formula for the sine of the sum of two angles is crucial. This formula is: \[ \sin(A+B) = \sin(A) \cos(B) + \cos(A) \sin(B) \] This identity decomposes the sine of a sum into products of sine and cosine of individual angles. This is incredibly useful because it translates an angle sum into a format involving the basic trigonometric functions, sine and cosine. By doing so, it allows us to substitute and manipulate the equation more easily, providing a path towards verifying more complex identities step by step. Understanding these formulas gives you the tools to simplify equations involving sums of angles. Whenever you deal with such problems, it’s crucial to remember and apply this sum identity.

Algebraic Manipulation

Algebraic manipulation involves systematically transforming equations or expressions to simpler, more readable forms, often enhancing understanding and providing paths to solutions. In the context of verifying trigonometric identities, it plays a significant role. When faced with complexities such as trigonometric identities, breaking down fractions or expressions into simpler parts through division, as seen in our original step-by-step solution, is a classic trick. For example, when we have the fraction: \[ \frac{ \sin(A) \cos(B) + \cos(A) \sin(B)}{\cos A \cos B} \] we split it into two separate fractions:

• \( \frac{ \sin(A) \cos(B)}{ \cos A \cos B} \)
• \( \frac{ \cos(A) \sin(B)}{ \cos A \cos B} \)

Breaking it down in such a way makes it easier to see how these can be further simplified using the definitions of the tangent function. Each fraction simplifies further to simpler trigonometric expressions, allowing you to confirm the identity with confidence. Algebraic manipulation is like laying out a puzzle – place each piece logically, and eventually, the complete picture (or solution) emerges.

Angle Sum Identity

The concept of the angle sum identity is essential when dealing with trigonometric identities involving sums of angles. It states how sine and cosine behave when applied to the sum of two angles, providing a way to express these sums in terms of products of single angles' sine and cosine. For sine, the identity is: \[ \sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B) \] And similarly, for cosine: \[ \cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B) \] These identities are like keys to solving trigonometric puzzles—they unlock the relationships between angles and enable simplification or verification of identities. In the exercise, using the angle sum identity allows you to substitute and transform the original equation step by step into a recognizable, simpler form involving tangent:

• \( \frac{ \sin(A + B)}{\cos A \cos B} = \tan(A) + \tan(B) \)

It beautifully showcases how breaking down an angle sum using the identity results in individual components that simplify into known trigonometric values like tangent.