Problem 32 - Algebra 2

Question

You can determine whether or not an equation may be a trigonometric identity by graphing the expressions on either side of the equals sign as two separate functions. If the graphs do not match, then the equation is not an identity. If the two graphs do coincide, the equation might be an identity. The equation has to be verified algebraically to ensure that it is an identity. $$ (1+\sin x)(1-\sin x)=\cos ^{2} x $$

Step-by-Step Solution

Answer

The equation is a trigonometric identity.

Step 1: Expand the Left-Hand Side

The left-hand side of the equation is \((1 + \sin x)(1 - \sin x)\). Use the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) to expand this:\[(1 + \sin x)(1 - \sin x) = 1^2 - (\sin x)^2 = 1 - \sin^2 x.\]

Step 2: Use a Trigonometric Identity

Recall the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\).Therefore, \(1 - \sin^2 x = \cos^2 x\).

Step 3: Compare Both Sides of the Equation

After simplification, both sides of the equation are: \(1 - \sin^2 x = \cos^2 x\).This confirms that both sides match, indicating that the original equation is an identity.

Key Concepts

Difference of Squares

The difference of squares is a foundational algebraic concept that often appears in trigonometry, particularly when simplifying expressions. The formula for the difference of squares states that \[(a + b)(a - b) = a^2 - b^2\]This principle is used to simplify or transform an expression. For beginners, it's important to note that the terms must be in the form of a sum and a difference to apply this identity.
In this specific trigonometric exercise, we use \[(1 + \sin x)(1 - \sin x)\]Here, it fits the difference of squares format because it is the product of two binomials: one is the sum and the other is the difference of the same terms (1 and \(\sin x\)). When applying the formula:

• The '1' term represents \(a\).
• The '\(\sin x\)' term represents \(b\).

This leads to:\[1^2 - (\sin x)^2 = 1 - \sin^2 x\]This transformation simplifies the expression, making it easier to work with in solving trigonometric equations.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry. This identity is derived from the Pythagorean theorem and is expressed as:\[\sin^2 x + \cos^2 x = 1\]This equation is immensely useful for transforming expressions, as it directly relates \(\sin\) and \(\cos\) in terms of the unit circle. Remember that the Pythagorean identity holds true for all values of \(x\).
In our exercise, we rearrange the identity to solve for \(\cos^2 x\):\[\cos^2 x = 1 - \sin^2 x\]By substituting \(1-\sin^2 x\) into the equation from the previous step, we confirm that both the transformed left side (\(1 - \sin^2 x\)) and the given right side (\(\cos^2 x\)) are equivalent. Hence, this substitution validates the expression as a trigonometric identity.
This identity not only simplifies expressions but also verifies that two different-looking expressions are indeed equal.

Graphical Verification of Trigonometric Identities

Graphical verification is a technique used alongside algebraic proofs to determine whether an equation holds true as a trigonometric identity. The primary method involves graphing each side of the equation separately. If the graphs coincide exactly over their domain, the two expressions are likely to be identities.
For the equation:\[(1 + \sin x)(1 - \sin x) = \cos^2 x\]1. Graph \(y = (1 + \sin x)(1 - \sin x)\) on one side.2. Graph \(y = \cos^2 x\) on the other.
Observing their graphs can visually reaffirm that they trace the same path and coincide, suggesting the two expressions are the same.This method acts as an initial check—it helps visually establish that the two forms are identical within a certain interval. However, graphical verification isn't foolproof; it must be complemented with algebraic methods like those mentioned above for complete validation. This combination ensures rigorous confirmation that an equation is indeed a trigonometric identity.