Problem 66 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\cot \theta+\tan \theta=\csc \theta \sec \theta$$

Step-by-Step Solution

Answer

The identity is verified: \( \cot \theta + \tan \theta = \csc \theta \sec \theta \).

Step 1: Express in terms of Sine and Cosine

Start by rewriting the trigonometric functions on the left-hand side in terms of sine and cosine. Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Then, the left side becomes:\[ \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \]

Step 2: Combine into a Single Fraction

To add these two fractions, find a common denominator. The common denominator for \( \sin \theta \) and \( \cos \theta \) is \( \sin \theta \cos \theta \). Rewrite the expression with this common denominator:\[ \frac{\cos \theta \cdot \cos \theta + \sin \theta \cdot \sin \theta}{\sin \theta \cos \theta} \]

Step 3: Simplify Numerator Using Pythagorean Identity

Notice that the numerator can be simplified using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \). Thus, the numerator becomes 1:\[ \frac{1}{\sin \theta \cos \theta} \]

Step 4: Separate into Products of Reciprocals

The expression \( \frac{1}{\sin \theta \cos \theta} \) can be rewritten as the product of reciprocals of \( \sin \theta \) and \( \cos \theta \), which gives:\[ \frac{1}{\sin \theta} \cdot \frac{1}{\cos \theta} \]

Step 5: Rewrite as Cosecant and Secant Functions

Recognize that \( \frac{1}{\sin \theta} = \csc \theta \) and \( \frac{1}{\cos \theta} = \sec \theta \). So, the expression simplifies to:\[ \csc \theta \sec \theta \]

Step 6: Conclusion: Verify Identity

You have successfully transformed the left-hand side \( \cot \theta + \tan \theta \) to match the right-hand side \( \csc \theta \sec \theta \). Therefore, the identity is verified as correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are crucial in mathematics for describing the relationships between the angles and sides of triangles. They are essential in various fields including physics, engineering, and computer science. The primary trigonometric functions are:

• Sine (\( \sin \theta \)),
• Cosine (\( \cos \theta \)),
• Tangent (\( \tan \theta \)),
• Cotangent (\( \cot \theta \)),
• Secant (\( \sec \theta \)),
• Cosecant (\( \csc \theta \)).

Each of these functions can express the ratio between different sides of a right triangle.
They allow us to translate angles into a usable form for equations and identities that describe rotational and wave phenomena in the real world.

Sine and Cosine

Sine and cosine are fundamental building blocks in trigonometry. These functions represent projections of a point rotating around a unit circle.
For an angle \( \theta \), the sine of the angle is the vertical coordinate, and the cosine is the horizontal coordinate when placed on the unit circle.

• The sine function is defined as: \[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]
• The cosine function is defined as:\[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \]

These functions are periodic, meaning they repeat values in a continuous wave-like pattern, with a period of \( 2\pi \) radians.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It relates the square of sine and cosine functions to a constant:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
This identity is derived from the Pythagorean theorem and is essential for simplifying expressions involving trigonometric functions.
For example, when transforming expressions like \( \cot \theta + \tan \theta \) into another form, the Pythagorean identity helps simplify the numerator by reducing terms to 1.
This simplification reveals deeper properties of trigonometric expressions and is foundational in solving trigonometric equations.

Cosecant and Secant

Cosecant and secant are the reciprocals of the sine and cosine functions respectively.

• Cosecant (\( \csc \theta \)) is defined as:\[ \csc \theta = \frac{1}{\sin \theta} \]
• Secant (\( \sec \theta \)) is defined as:\[ \sec \theta = \frac{1}{\cos \theta} \]

These functions are particularly useful when dealing with inverse ratios in a trigonometric context.
They also often appear in verifying or transforming trigonometric identities like the one seen in the original exercise.
Utilizing these functions allows us to express expressions cleanly and concisely, highlighting the symmetry and relationships within a trigonometric identity.

Common Denominator

Finding a common denominator is essential when adding fractions, especially in trigonometric identities. In trigonometry, the denominators come from functions like sine and cosine.
For example, when adding \( \frac{\cos \theta}{\sin \theta} \) and \( \frac{\sin \theta}{\cos \theta} \), the common denominator is \( \sin \theta \cos \theta \).

• This allows us to rewrite the fractions with a shared base so they can be added together: \[ \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} \]
• Using the Pythagorean identity simplifies the numerator.

Understanding how to work with common denominators in trigonometric expressions is vital for simplifying and solving complex identities.