Problem 111 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{2\left(\sin x-\sin ^{3} x\right)}{\cos x}=\sin 2 x$$

Step-by-Step Solution

Answer

The equation \( \frac{2(\sin x - \sin^3 x)}{\cos x} = \sin 2x \) is an identity.

Step 1: Simplify the Left Side

Start by simplifying the left side of the equation: \( \frac{2(\sin x - \sin^3 x)}{\cos x} \). Notice that \( \sin^3 x \) can be factored from \( \sin x \), resulting in \( \sin x (1 - \sin^2 x) \). Utilize the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to express \( 1 - \sin^2 x \) as \( \cos^2 x \). This gives: \( \frac{2(\sin x \cdot \cos^2 x)}{\cos x} \).

Step 2: Cancel \( \cos x \)

Next, cancel one \( \cos x \) from the numerator with the \( \cos x \) in the denominator. This simplifies to \( 2 \sin x \cos x \).

Step 3: Use Double Angle Identity

Recognize that \( 2 \sin x \cos x \) is the double angle identity for sine, \( \sin 2x \). Thus, \( 2 \sin x \cos x = \sin 2x \).

Step 4: Conclusion: Verify the Identity

Both the simplified left side \( 2 \sin x \cos x \) and the right side \( \sin 2x \) of the equation are identical, confirming the given equation as an identity: \( \frac{2(\sin x - \sin^3 x)}{\cos x} = \sin 2x \).

Key Concepts

Double Angle Identity

The double angle identity is a crucial concept in trigonometry that allows us to express trigonometric functions of double angles in terms of single angles.
It's especially handy for simplification and solving trigonometric equations.
For the sine function, the double angle identity is given by:\[\sin 2x = 2 \sin x \cos x\]This identity reveals how the sine of a double angle (2x) can be broken down into the components of the sine and cosine of the single angle (x).
The given exercise provides a perfect example of how this identity is used:
We recognize \(2 \sin x \cos x\) as \(\sin 2x\), simplifying our calculations significantly. Remember:

• Double angle identities work not just for sine but also for cosine and tangent.
• They are derived from basic trigonometric identities and are foundational for calculus and advanced mathematics.
• Make sure to practice these identities for mastery, as they are commonly used in exams.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry. It connects the sine and cosine of an angle, based on the Pythagorean theorem.
The identity is expressed as:\[\sin^2 x + \cos^2 x = 1\]This identity is extremely useful for simplifying expressions and solving equations in trigonometry.
In the exercise, it allows us to rewrite \(1 - \sin^2 x\) as \(\cos^2 x\).
This kind of transformation is crucial, making it easier to recognize known identities or reach a desired form. Essential points:

• The Pythagorean identity forms the basis for many other trigonometric identities.
• It provides a quick check to ensure computations involving sine and cosine are consistent.
• Ensure you can flexibly manipulate this identity for both simplification and verification tasks.

Trigonometric Simplification

Trigonometric simplification is a process used to make complex trigonometric expressions more manageable and easier to work with.
Mastering this skill helps in solving equations and proving identities efficiently. In the given exercise, simplification involved several steps:

• Factoring \(\sin^3 x\) to \(\sin x (1 - \sin^2 x)\).
• Replacing \(1 - \sin^2 x\) with \(\cos^2 x\) using the Pythagorean identity.
• Cancelling out \(\cos x\) from the numerator and the denominator.
• Recognizing the form \(2 \sin x \cos x\) as \(\sin 2x\).

Key takeaways for simplification:

• Always look for opportunities to factorize expressions.
• Use known identities to transform parts of the expression into more recognizable forms.
• Perform cancellation cautiously to avoid errors, ensuring terms in the denominator are not zero to preserve the expression's validity.
• Practice regularly to recognize patterns and common forms quickly.