Problem 43 - Precalculus: Functions and Graphs

Question

Exer. \(37-46:\) Verify the identity. $$\sin (u+v) \cdot \sin (u-v)=\sin ^{2} u-\sin ^{2} v$$

Step-by-Step Solution

Answer

The identity is verified by expanding and simplifying using identities and algebra.

Step 1: Use Sum and Difference Identities

Recall the sum and difference identities for sine. These are: \[ \sin(u+v) = \sin u \cos v + \cos u \sin v \]\[ \sin(u-v) = \sin u \cos v - \cos u \sin v \]

Step 2: Apply Sum Identity for Sine Scenarios

Apply the identity for sine of \((u+v)\) and \((u-v)\) separately: \[ \sin(u+v) = \sin u \cos v + \cos u \sin v \] \[ \sin(u-v) = \sin u \cos v - \cos u \sin v \]

Step 3: Multiply Sine Expressions

The left side of the identity becomes: \[(\sin u \cos v + \cos u \sin v)(\sin u \cos v - \cos u \sin v)\] Expand this product using the formula \((a+b)(a-b) = a^2 - b^2\).

Step 4: Apply the Difference of Squares Formula

Applying \((a+b)(a-b) = a^2 - b^2\), where \(a = \sin u \cos v\) and \(b = \cos u \sin v\), results in:\[(\sin u \cos v)^2 - (\cos u \sin v)^2\]

Step 5: Simplify Each Square

Simplify each term:\[ (\sin u \cos v)^2 = \sin^2 u \cos^2 v \]\[ (\cos u \sin v)^2 = \cos^2 u \sin^2 v \]

Step 6: Use the Pythagorean Identity

Recall the Pythagorean identity: \[ \sin^2 u + \cos^2 u = 1 \] \[ \sin^2 v + \cos^2 v = 1 \] Substitute to simplify terms inside the expression:

Step 7: Compare Both Sides

After simplification, you obtain: \[ \sin^2 u \cos^2 v - \cos^2 u \sin^2 v \]Rewrite this equivalent to \[ \sin^2 u - \sin^2 v \]Both sides match, confirming the identity.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It helps us understand the relationship between angles and side lengths in a triangle. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. For example, if we have a triangle with an angle \( \theta \), the sine function is expressed as:\[ \sin \theta = \frac{\text{length of opposite side}}{\text{length of hypotenuse}} \]In trigonometric identities, the sine function helps us express other trigonometric functions and prove identities. The problem highlights two particular identities: the sum and difference formulas for sine. These are useful for breaking down complex trigonometric expressions. Here's how they work:

• \( \sin(u+v) = \sin u \cos v + \cos u \sin v \)
• \( \sin(u-v) = \sin u \cos v - \cos u \sin v \)

Using these identities, we can multiply and manipulate trigonometric expressions more easily. This manipulation often leads to simplifying or verifying an identity, as in the given exercise.

Difference of Squares

The difference of squares is a mathematical formula used to simplify expressions where two squared terms are subtracted. This useful algebraic identity states:\[ a^2 - b^2 = (a+b)(a-b) \]In the context of trigonometric identities, this rule simplifies the multiplication of angles' sine expressions. The problem uses the identity:\[ (\sin u \cos v + \cos u \sin v)(\sin u \cos v - \cos u \sin v) \]By identifying terms \( a = \sin u \cos v \) and \( b = \cos u \sin v \), the formula becomes:\[ (\sin u \cos v)^2 - (\cos u \sin v)^2 \]Applying the difference of squares concept simplifies the expression. This simplification is crucial for verifying the given trigonometric identity, ensuring that both sides of the equation are equal.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry that relates the squares of the sine and cosine of the same angle. The identity is written as:\[ \sin^2 u + \cos^2 u = 1 \]This relationship arises from the Pythagorean theorem, which defines the connection between the sides of a right triangle. By using this identity, we can substitute and simplify expressions within trigonometric equations.For example, if we need to express \( \cos^2 u \) in terms of \( \sin^2 u \), we simply rearrange the Pythagorean identity:\[ \cos^2 u = 1 - \sin^2 u \]In the trigonometric identity verification given in the exercise, the Pythagorean identity was crucial for simplifying terms like \( \sin^2 u \cos^2 v - \cos^2 u \sin^2 v \) to show equivalence between both sides. Such simplifications ensure the original trigonometric identity is valid, using the fundamental properties of sine and cosine.