Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{\cos \theta+1}{\tan ^{2} \theta}=\frac{\cos \theta}{\sec \theta-1}\)

Step-by-Step Solution

Answer

The equation is an identity as both sides simplify to the same expression \( \frac{\cos^2 \theta}{1 - \cos \theta} \).

Step 1: Simplifying the Left Side

Start by rewriting \( \tan^2 \theta \) as \( \frac{\sin^2 \theta}{\cos^2 \theta} \). This gives us the left side of the equation as \( \frac{\cos \theta + 1}{\tan^2 \theta} = \frac{\cos \theta + 1}{\frac{\sin^2 \theta}{\cos^2 \theta}} = \frac{(\cos \theta + 1) \cos^2 \theta}{\sin^2 \theta} \).

Step 2: Simplifying the Right Side

Now, simplify the right side of the equation. Rewrite \( \sec \theta \) as \( \frac{1}{\cos \theta} \) to get \( \frac{\cos \theta}{\sec \theta - 1} = \frac{\cos \theta}{\frac{1}{\cos \theta} - 1} = \frac{\cos^2 \theta}{1 - \cos \theta} \).

Step 3: Finding a Common Expression

Both sides should simplify to the same expression. Return to the left side and attempt to manipulate it into the form on the right side. Multiply top and bottom by \( 1 + \cos \theta \), a form often used to rationalize expressions, to obtain \( \frac{(\cos \theta+1)^2 \cos^2 \theta}{\sin^2 \theta} \). Since \( \sin^2 \theta = 1 - \cos^2 \theta \), we rewrite the denominator using the identity \( 1 - \cos^2 \theta = (1 - \cos \theta)(1 + \cos \theta) \).

Step 4: Verify the Simplification

The expression on the left after simplifying and using identities is \( \frac{(\cos \theta + 1) \cos^2 \theta}{(1 - \cos \theta)(1 + \cos \theta)} \), which further simplifies to \( \frac{\cos^2 \theta}{1 - \cos \theta} \). This is the same expression as obtained for the right side.

Step 5: Conclusion

Since the expressions for both the left and the right sides of the equation simplify to the same form \( \frac{\cos^2 \theta}{1 - \cos \theta} \), we have verified that the equation is an identity.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a more manageable form. This often requires using basic trigonometric identities. These identities allow us to substitute one function for another, simplifying our work. Consider the original expression \( \frac{\cos \theta+1}{\tan ^{2} \theta} \). Here, \( \tan^2 \theta \) is rewritten as \( \frac{\sin^2 \theta}{\cos^2 \theta} \).
This substitution simplifies the expression to \( \frac{\cos \theta + 1}{\frac{\sin^2 \theta}{\cos^2 \theta}} \), further breaking it down to \( \frac{(\cos \theta + 1) \cos^2 \theta}{\sin^2 \theta} \).
It becomes crucial to have a clear understanding of these identities, as they form the cornerstone for simplifying complex trigonometric expressions effectively.

Verify Trigonometric Identities

Verifying trigonometric identities means proving that two different expressions are equivalent. In this exercise, we aim to show that both sides of the equation have the same value no matter what angle \( \theta \) is.
To verify, we simplify each side independently, using identities. On the right side, \( \sec \theta \) is replaced with \( \frac{1}{\cos \theta} \), transforming the expression to \( \frac{\cos \theta}{\frac{1}{\cos \theta} - 1} \).
Simplifying this results in \( \frac{\cos^2 \theta}{1 - \cos \theta} \).
After simplification, both sides equate to \( \frac{\cos^2 \theta}{1 - \cos \theta} \), verifying the identity. This process shows not only the equivalency but also deepens our understanding of trigonometric relationships.

Trigonometric Functions

Trigonometric functions, such as \( \sin \theta \), \( \cos \theta \), \( \tan \theta \), and \( \sec \theta \), play vital roles in a variety of mathematical problems. They are functions of an angle and provide the foundation for working with trigonometric identities.
For example, the trigonometric function \( \tan \theta \) is defined as the ratio \( \frac{\sin \theta}{\cos \theta} \), while \( \sec \theta \) is \( \frac{1}{\cos \theta} \).
These simple relationships can be the key to solving more complex equations and understanding trigonometric identities.

• \( \sin^2 \theta + \cos^2 \theta = 1 \) is one key identity often used to convert expressions.
• Understanding how to manipulate and apply these identities and functions can simplify problem solving significantly.

By mastering these basic functions and identities, you can unlock solutions to problems that initially seem difficult.