Problem 23 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin 3 x-\sin x}{\cos 3 x-\cos x}=-\cot 2 x$$

Step-by-Step Solution

Answer

The identity \( \frac{\sin 3 x-\sin x}{\cos 3 x-\cos x} = -\cot 2 x \) is verified.

Step 1: Express the subtraction of sines and cosines using formulas

Recall that there are formulas to express the subtraction of two sine or cosine functions: \n\n \( \sin A - \sin B = 2 \cos \frac{A+B}{2} \sin \frac{A-B}{2} \)\n \( \cos A - \cos B = -2 \sin \frac{A+B}{2} \sin \frac{A-B}{2} \)\n\nAppling these formulas to \(\sin 3x - \sin x\) and \(\cos 3x - \cos x\) results in:\n\n \( \frac{2 \cos 2x \sin x}{-2 \sin 2x \sin x} \)

Step 2: Simplify the obtained expression

Having similar terms in the numerator and denominator enables us to simplify the expression. Cancellation gives: \(-\cot 2x\)

Step 3: Conclusion

The expression on the left side has been transformed to the form on the right side. Thus, the trigonometric identity is verified.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding relationships between the angles and sides of triangles, especially right-angled triangles. The primary trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function represents a specific ratio:

• The sine of an angle is the ratio of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the opposite side to the adjacent side.

Understanding these functions helps in analyzing and solving more complex identities. Using these trigonometric functions, we can express other functions like cotangent, secant, and cosecant:

• Cotangent (\( \cot \theta \)) is the reciprocal of the tangent, or \( \frac{1}{\tan \theta} \).
• Secant (\( \sec \theta \)) is the reciprocal of cosine, or \( \frac{1}{\cos \theta} \).
• Cosecant (\( \csc \theta \)) is the reciprocal of sine, or \( \frac{1}{\sin \theta} \).

An understanding of these relationships is crucial when handling trigonometric identities and expressions.

Angle Sum and Difference Formulas

The angle sum and difference formulas are tools in trigonometry that help derive the sine, cosine, or tangent of sum or differences of angles. They can transform expressions by breaking complex angles into smaller, more manageable components.
These formulas are extensively used for simplifying trigonometric identities, solving equations, and simplifying expressions in geometry and calculus.

The key angle difference formulas for sine and cosine are:

• Sine difference: \[ \sin A - \sin B = 2 \cos \frac{A+B}{2} \sin \frac{A-B}{2} \]
• Cosine difference: \[ \cos A - \cos B = -2 \sin \frac{A+B}{2} \sin \frac{A-B}{2} \]

By applying these formulas, complex trigonometric expressions can be simplified to simpler ratios, as demonstrated in the exercise where \( \sin 3x - \sin x \) and \( \cos 3x - \cos x \) were broken down to facilitate solving the identity.

Simplifying Expressions

Simplifying expressions is a fundamental step in solving mathematical problems, including verifying trigonometric identities. By breaking down complex expressions into simpler components, calculations become more straightforward and manageable.

Here's a basic approach to simplifying expressions:

• Identify common factors or terms that appear in both the numerator and denominator and cancel them out if possible.
• Use trigonometric identities, such as reciprocal identities, Pythagorean identities, or angle sum and difference formulas to reduce complexity.
• Always aim for a straightforward yet accurate representation of the expression.

In the given exercise, simplifying involved using the formulas for differences in sine and cosine to create a more accessible expression. It allowed the direct application of simplifying steps by cancelling out similar terms, ultimately revealing the trigonometric identity \(-\cot 2x\).
Proficiency in these techniques not only aids in handling trigonometric identities but also in other mathematical disciplines, improving problem-solving skills.