Problem 29 - Precalculus

Question

Verify the identity. $$\cos x-\cos ^{3} x=\cos x \sin ^{2} x$$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \cos x \sin^2 x \).

Step 1: Understand the Expression and the Identity

The expression given is \( \cos x - \cos^3 x \) which we need to verify is equal to the identity \( \cos x \sin^2 x \). Recognize that both sides involve \( \cos x \) as a common factor.

Step 2: Factor the Left Side of the Equation

Observe that \( \cos x - \cos^3 x \) can be factored. Use the formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) for factoring, though here use simpler: \( \cos x (1 - \cos^2 x) \).

Step 3: Use the Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 x = 1 - \cos^2 x \). Substitute \( 1 - \cos^2 x \) with \( \sin^2 x \) in the factored form.

Step 4: Simplify the Factored Expression

Substitute \( \sin^2 x \) for \( 1 - \cos^2 x \) in \( \cos x (1 - \cos^2 x) \). This gives \( \cos x \sin^2 x \).

Step 5: Verify the Identity

Both sides are now \( \cos x \sin^2 x \), confirming that the original expression is equivalent to the identity provided.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It connects the sine and cosine functions through the equation \( \sin^2 x + \cos^2 x = 1 \). This identity is derived from the Pythagorean theorem, which deals with right triangles. Here, the "1" represents the hypotenuse's length in a unit circle, and \( \sin^2 x \) and \( \cos^2 x \) are the lengths of the opposite and adjacent sides, respectively.
This identity is useful when solving trigonometric equations or verifying trigonometric expressions. For example, we use this identity to transform expressions involving \( \cos^2 x \) or \( \sin^2 x \) into each other. In our example, we replaced \( 1 - \cos^2 x \) with \( \sin^2 x \) using this identity to simplify the expression. Remember, knowing the Pythagorean identity is invaluable for understanding the relationship between sine and cosine functions.

cosine function

The cosine function, denoted as \( \cos x \), is one of the primary trigonometric functions. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function is periodic with a period of \( 2\pi \). It belongs to the family of even functions, meaning \( \cos(-x) = \cos x \).
The graph of \( \cos x \) is a wave that starts at 1, dips to -1, and returns, reflecting symmetrical properties along the y-axis. The cosine function is essential in defining the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Understanding how \( \cos x \) behaves is crucial when verifying identities or calculating angles. In the provided exercise, \( \cos x \) was factored out, showing its pivotal role in simplifying expressions.

sine function

The sine function, \( \sin x \), is another core trigonometric function, defined as the ratio of the opposite side to the hypotenuse in a right triangle. It complements the cosine function and also has a period of \( 2\pi \). The sine function is odd, meaning \( \sin(-x) = -\sin x \), showing that its graph is symmetric with respect to the origin.
In its graph, \( \sin x \) starts at 0, reaches 1, dips to -1, and back to 0 over a \( 2\pi \) interval. This function is integral in the identity \( \sin^2 x + \cos^2 x = 1 \), as seen in the exercise where \( \sin^2 x \) was substituted for \( 1 - \cos^2 x \). Comprehending the sine function aids in manipulating and transforming trigonometric expressions.

factoring expressions

Factoring expressions involves rewriting a polynomial or expression as a product of its factors. It's a crucial skill in algebra that simplifies calculations and verifies identities. For our trigonometric example, we factored \( \cos x - \cos^3 x \) by recognizing \( \cos x \) as a common factor, writing it as \( \cos x (1 - \cos^2 x) \).
This technique is not limited to algebraic expressions, but extends to trigonometric expressions. Factoring can often reveal hidden relationships or allow for substitution, as seen when we applied the Pythagorean identity. Practicing factoring helps in solving complex problems and simplifies trigonometric proofs, making it an essential mathematical technique to master.