Problem 106 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Show that \(\cos ^{3} x=\cos x-\sin ^{2} x \cos x\).

Step-by-Step Solution

Answer

\cos^{3}x = \cos x \ - \ \sin^{2} x \cos x.

Step 1: Start with the given identity

We need to show that \ \ \ \ \ \ \ \( \cos ^{3} x=\cos x-\sin ^{2} x \cos x \).

Step 2: Break down the right side

Rewrite the right side of the equation: \ \ \ \ \ \ \ \( \cos x-\sin ^{2} x \cos x \ =\cos x(1-\sin ^{2} x) \).

Step 3: Simplify using Pythagorean Identity

Recall the Pythagorean Identity: \ \ \ \ \ \ \ \( 1 =\sin^{2} x +\cos^{2} x \). Substitute \( 1 -\sin^{2} x \) with \( \cos^{2} x. \Then, \cos x(1-\sin ^{2} x) =\cos x (\cos ^{2} x). \)

Step 4: Verify the final expression

Multiply \ \ \ \ \cos x \ times \ \cos^{2} x, resulting in \ \ \ \( \cos^{3} x = \cos x \cdot \cos^2 x. \). This confirms the original given identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental trigonometric identities. It states that for any angle \(x\), the relationship between the sine and cosine functions is described by:
\[ \text {sin}^2 x + \text {cos}^2 x = 1 \] This means that no matter the value of \(x\), the sum of the squares of the sine and cosine functions will always equal 1.

In our exercise, we used this identity to simplify the given equation. By expressing \(1 - \text {sin}^2 x\) in terms of \( \text {cos}^2 x\), we made it easier to show that \( \text {cos}^3 x = \text {cos} x - \text {sin}^2 x \text {cos} x \).

Trigonometric Simplification

Trigonometric simplification is the process of reducing trigonometric expressions to simpler forms. This often involves using fundamental identities like the Pythagorean Identity.

In our example, we simplified \( \text {cos} x - \text {sin}^2 x \text {cos} x\) by factoring out \( \text {cos} x\), giving us:
\( \text {cos} x(1 - \text {sin}^2 x) \).

Next, we used the Pythagorean Identity to replace \(1 - \text {sin}^2 x\) with \( \text {cos}^2 x\), turning our expression into:
\(\text {cos} x \text {cos}^2 x)\).
This simplifies to \( \text {cos}^3 x\), matching the given identity.

Cosine Function

The cosine function, often denoted as \( \text {cos} x \), is one of the principal trigonometric functions. It represents the adjacent side over the hypotenuse in a right-angled triangle.

The cosine function also has several important properties and identities linked to it. For instance, \( \text {cos}^2 x = 1 - \text {sin}^2 x\), which is fundamental to many trigonometric simplifications.

In our problem, understanding the cosine function and its properties helped us simplify \( \text {cos} x - \text {sin}^2 x \text {cos} x\) to \( \text {cos} ^{3} x \). Using properties like these makes solving trigonometric identities more straightforward.