Problem 63 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec \theta-\cos \theta=\tan \theta \sin \theta$$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Understand the Identity

We need to verify the identity \( \sec \theta - \cos \theta = \tan \theta \sin \theta \) by transforming the left-hand side into the right-hand side.

Step 2: Express \(\sec \theta \) in terms of \(\cos \theta \)

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this expression into the left-hand side to get:\[ \frac{1}{\cos \theta} - \cos \theta \]

Step 3: Combine the Terms on the Left-Hand Side

To combine the terms, find a common denominator. The common denominator for \( \frac{1}{\cos \theta} \) and \( \cos \theta \) is \( \cos \theta \).Write the expression as:\[ \frac{1 - \cos^2 \theta}{\cos \theta} \]

Step 4: Simplify using Trigonometric Identity

Use the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \). Substitute \( 1 - \cos^2 \theta \) with \( \sin^2 \theta \):\[ \frac{\sin^2 \theta}{\cos \theta} \]

Step 5: Simplify the Expression

The expression \( \frac{\sin^2 \theta}{\cos \theta} \) can be split as:\[ \sin \theta \times \frac{\sin \theta}{\cos \theta} \]Recognize that \( \frac{\sin \theta}{\cos \theta} = \tan \theta \), so:\[ \sin \theta \tan \theta \]

Step 6: Verify the Identity

We have transformed the left-hand side into \( \sin \theta \tan \theta \), which is exactly the right-hand side of the original identity:\[ \tan \theta \sin \theta \] Thus, the identity is verified.

Key Concepts

Secant Function

The secant function, represented as \( \sec \theta \), is an essential trigonometric function. It's closely related to the cosine function. In fact, the secant function is the reciprocal of the cosine function.

This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Understanding this relationship helps simplify and transform trigonometric expressions. Whenever you see \( \sec \theta \), you can replace it with \( \frac{1}{\cos \theta} \) to make calculations easier.

This transformation is particularly useful when verifying identities or solving equations. By expressing \( \sec \theta \) in terms of \( \cos \theta \), a clearer association with other trigonometric functions can be seen, helping to solve problems like \( \sec \theta - \cos \theta = \tan \theta \sin \theta \).

Tangent Function

The tangent function, denoted as \( \tan \theta \), is another fundamental trigonometric function.

It is defined as the ratio of the sine function to the cosine function:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This function is invaluable in various mathematical problems involving right triangles as it relates the angle to the opposite and adjacent sides.

In problems like verifying the given identity \( \sec \theta - \cos \theta = \tan \theta \sin \theta \), recognizing \( \tan \theta \) as \( \frac{\sin \theta}{\cos \theta} \) allows for rearranging and simplifying expressions. By expressing \( \tan \theta \) in terms of \( \sin \theta \) and \( \cos \theta \), the complexity of an equation often reduces, making it simpler to verify identities.

Pythagorean Identity

One of the most powerful tools in trigonometry is the Pythagorean identity. This identity relates the square of the sine and cosine functions:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This fundamental equation is the basis for deriving other useful identities. For example, you can rearrange this identity to express \( \sin^2 \theta \) as:

• \( \sin^2 \theta = 1 - \cos^2 \theta \)

This rearrangement is particularly handy in transforming trigonometric expressions, as seen in the step-by-step solution to verify the identity \( \sec \theta - \cos \theta = \tan \theta \sin \theta \).

By using the Pythagorean identity, expressions like \( 1 - \cos^2 \theta \) can be replaced with \( \sin^2 \theta \), simplifying the expression and making the verification process more straightforward. Mastering the Pythagorean identity is thus crucial for anyone looking to excel in trigonometry.