Problem 68 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \sin(x + y) \sin(x - y) = \sin^2 x - \sin^2 y \)

Step-by-Step Solution

Answer

After applying the sum and difference formulas for sine, expanding and using the Pythagorean identity, you can simplify the left side of the equation to match the right side, proving the identity \( \sin(x + y) \sin(x - y) = \sin^2 x - \sin^2 y \).

Step 1: Apply the sum and difference formulas

We know that \( \sin(a + b) = \sin a \cos b + \sin b \cos a \) and \( \sin(a - b) = \sin a \cos b - \sin b \cos a \). We apply these formulas to \( \sin(x + y) \) and \( \sin(x - y) \), giving us \((\sin x \cos y + \sin y \cos x) (\sin x \cos y - \sin y \cos x)\).

Step 2: Expand the expression

By expanding this expression we get \( (\sin^2 x \cos^2 y - \sin^2 y \cos^2 x) \) .

Step 3: Use Pythagorean identity

Using the Pythagorean identity \( \sin^2 a + \cos^2 a = 1 \), we can replace \( \cos^2 y \) in the first term with \( 1 - \sin^2 y \), and \( \cos^2 x \) in the second term with \( 1 - \sin^2 x \) obtaining \( \sin^2 x (1 - \sin^2 y) - \sin^2 y (1 - \sin^2 x) \)

Step 4: Simplify the expression

Finally, by simplifying this expression, we arrive at \( \sin^2 x - \sin^2 y \) which is the right hand side of the original identity, thereby proving the identity.

Key Concepts

Sum and Difference Formulas

Understanding the sum and difference formulas is crucial for proving various trigonometric identities. These formulas allow us to express the sine, cosine, or tangent of a sum or difference of angles in terms of the products of trigonometric functions of the individual angles.

For instance, the sum formula for sine is represented as:
\[ \text{For angles } a \text{ and } b,\]
\[ \text{Sin}(a + b) = \text{Sin}(a)\text{Cos}(b) + \text{Cos}(a)\text{Sin}(b)\]
Similarly, the difference formula for sine can be expressed as:
\[ \text{Sin}(a - b) = \text{Sin}(a)\text{Cos}(b) - \text{Cos}(a)\text{Sin}(b)\]
These formulas are immensely valuable when solving trigonometric equations or proving identities, as they deconstruct complex expressions into more manageable parts. In the given exercise, we applied these formulas to rewrite \( \text{Sin}(x + y) \) and \( \text{Sin}(x - y) \) in terms of \( \text{Sin}(x) \), \( \text{Cos}(y) \), \( \text{Sin}(y) \), and \( \text{Cos}(x) \), which laid the foundation for further simplifications.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the trigonometric functions sine and cosine. It's derived from the Pythagorean theorem, related to right-angled triangles, and is stated as:
\[ \text{For any angle } a,\]
\[ \text{Sin}^{2}(a) + \text{Cos}^{2}(a) = 1\]
This identity expresses that the sum of the squares of sine and cosine of an angle equals one. It enables us to transform trigonometric expressions and is especially handy when we need to convert between \( \text{Sin}^{2}(a) \) and \( \text{Cos}^{2}(a) \). In cumbersome trigonometric proofs, like in our exercise, the Pythagorean identity helps isolate a single trigonometric function by replacing its counterpart. For example, we replaced \( \text{Cos}^{2}(y) \) with \( 1 - \text{Sin}^{2}(y) \), and \( \text{Cos}^{2}(x) \) with \( 1 - \text{Sin}^{2}(x) \), a strategic move that simplified the expression efficiently towards the desired identity.

Trigonometric Functions

The trigonometric functions—sine (sin), cosine (cos), and tangent (tan), among others—are the backbone of trigonometry. They relate the angles of a triangle to the lengths of its sides. In a unit circle context, these functions acquire an even broader significance, allowing the calculation of angles and distances in a variety of settings.

Sine and cosine, the primary trigonometric functions used in our exercise, are defined for an angle, usually in a right-angled triangle or a unit circle. The definitions are:

• The sine of an angle in a right-angle triangle is the ratio between the opposite side and the hypotenuse.
• The cosine of an angle is the ratio between the adjacent side and the hypotenuse.

In the unit circle, these functions correspond to the y-value and x-value, respectively, of a point on the circle's circumference with a radius of one. Understanding these functions and their relationships—such as the one expressed by the Pythagorean identity—is not just a pillar of trigonometry but also a necessity for higher-level math and physics studies. In the context of our exercise, recognizing the specific properties of these functions allowed us to successfully manipulate the expressions and prove the trigonometric identity.