Problem 37 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( (1 + \sin y) [1 + \sin (-y)] = \cos^2 y \)

Step-by-Step Solution

Answer

Yes, the identity \( (1 + \sin y) [1 + \sin (-y)] = \cos^2 y \) is verified.

Step 1: Identify negative angle identity

First, recognize that \(\sin(-y) = -\sin y\). This is because sine function is an odd function.

Step 2: Simplification

Replace \(\sin(-y)\) in left side of equation with \(-\sin y\), so \([1 + \sin y][1 + \sin(-y)]\) becomes \([1 + \sin y][1 - \sin y]\).

Step 3: Expansion

Expand \([1 + \sin y][1 - \sin y]\), using the pattern that \((a+b)(a-b)=a^2 - b^2\), obtaining \(1 - \sin^2 y\).

Step 4: Use trigonometric identity

Finally, use the Pythagorean identity \(\sin^2 y + \cos^2 y = 1\). From here, you can solve for \(1 - \sin^2 y\), which equals to \(\cos^2 y\). That's exactly what the right side of the original equation is.

Key Concepts

Negative Angle Identity

Trigonometric identities help us understand the symmetrical properties of trigonometric functions. When we talk about the Negative Angle Identity for sine, it means acknowledging how the sine function behaves when its input is a negative angle.
For sine, this can be expressed as: \( \sin(-y) = -\sin(y) \). This identity is crucial because it simplifies expressions involving negative angles by converting them into simple negative terms of their positive counterparts. In our exercise, recognizing that \( \sin(-y) = -\sin(y) \) allows us to substitute \( -\sin y \) in the equation, shaping it into a more workable form.
This substitution is the starting point to apply other identities like the Pythagorean Identity in subsequent steps.

Pythagorean Identity

The Pythagorean Identity is derived from the Pythagorean theorem, providing a foundational relation involving sine and cosine functions: \( \sin^2 y + \cos^2 y = 1 \).
This identity is a powerful tool as it connects these two trigonometric functions. It allows us to express the square of one function in terms of the other. For instance, if you know \( \sin^2 y \), you can easily find \( \cos^2 y \) by rearranging the identity to: \( \cos^2 y = 1 - \sin^2 y \).
In solving our problem, this rearranged form of the identity helps us verify the original equation. By showing that \( 1 - \sin^2 y = \cos^2 y \), we confirm that we reach the equation's right-hand side, solidifying our solution.

Odd Function

Understanding the concept of odd functions is essential when dealing with trigonometric functions. A function is classified as odd if it satisfies \( f(-x) = -f(x) \). This characteristic demonstrates a form of symmetry about the origin on a graph.
The sine function is a perfect example of an odd function. This means that for any angle \( y \), \( \sin(-y) = -\sin(y) \).
This property is particularly useful when working with expressions like in our exercise because it helps simplify complex trigonometric equations by straightforward substitutions. By applying this property to negative angle identities, you gain the ability to transform difficult angles into manageable ones, setting the stage for further simplifications with other identities.