Problem 101 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Simplify the Left Side

Start by simplifying the left side of the identity, \(\frac{\tan ^{2} t-1}{\sec ^{2} t}\). Recall the Pythagorean identity \(\tan^2 t + 1 = \sec^2 t\), which implies \(\tan^2 t = \sec^2 t - 1\). Substitute this into the numerator: \(\tan^2 t - 1 = (\sec^2 t - 1) - 1 = \sec^2 t - 2\). Now the left side becomes \(\frac{\sec^2 t - 2}{\sec^2 t}\). This simplifies to \(1 - \frac{2}{\sec^2 t}\), which can be further simplified using the identity \(\sec^2 t = 1 + \tan^2 t\), leading to \(1 - \frac{2}{1 + \tan^2 t}\).

Step 2: Simplify the Right Side

For the right side \(\frac{\tan t-\cot t}{\tan t+\cot t}\), express \(\cot t\) as \(\frac{1}{\tan t}\). This gives \(\frac{\tan t - \frac{1}{\tan t}}{\tan t + \frac{1}{\tan t}}\). Simplify both the numerator and denominator by combining terms: The numerator simplifies to \(\frac{\tan^2 t - 1}{\tan t}\), and the denominator simplifies to \(\frac{\tan^2 t + 1}{\tan t}\). By dividing both by \(\tan t\), we get \(\frac{\tan^2 t - 1}{\tan^2 t + 1}\).

Step 3: Verify Identity

Compare both simplified sides. On the left, we have \(1 - \frac{2}{1 + \tan^2 t}\), which we can write as \(\frac{\tan^2 t - 1}{\tan^2 t + 1}\) using the previous simplification steps. The right side is already in this form: \(\frac{\tan^2 t - 1}{\tan^2 t + 1}\). Since they match, the equation \(\frac{\tan^2 t - 1}{\sec^2 t} = \frac{\tan t - \cot t}{\tan t + \cot t}\) is indeed an identity.

Key Concepts

Pythagorean Identities

In trigonometry, Pythagorean identities are a cornerstone concept. These identities are derived from the Pythagorean theorem, which relates the three sides of a right triangle. There are three main Pythagorean identities in trigonometry:

• The classic one: \( \sin^2\theta + \cos^2\theta = 1 \)
• The second identity, closely related, is \( 1 + \tan^2\theta = \sec^2\theta \)
• And the third identity is \( 1 + \cot^2\theta = \csc^2\theta \)

These identities are tremendously useful when simplifying or transforming trigonometric expressions. In the given exercise, we particularly use the identity \( 1 + \tan^2 t = \sec^2 t \). By substituting this identity into the expression, we can transform and simplify trigonometric equations, leading us to prove or verify the statements.

Simplifying Expressions

Simplifying trigonometric expressions can greatly help in understanding and solving problems. This involves reducing an expression to its simplest form, often using identities to substitute and combine terms.
In the exercise, we have a complex expression that seems overwhelming at first. The key is breaking it down with the aid of identities. Consider using the identities to replace terms, and convert everything into functions like sine, cosine, or tangent.
Applying the Pythagorean identity \( \tan^2 t = \sec^2 t - 1 \) transforms and reduces the complexity. Simplifying further to exchange terms like \( \sec^2 t \) into equivalent expressions naturally reduces clutter and helps in easier manipulation.
For any learner, practice simplifying trigonometric expressions by targeting fraction reduction, using core identities, and seeking terms that can cancel out is highly beneficial. This method often unveils a clearer picture of the equation.

Verifying Identities

Verifying trigonometric identities is a systematic process of proving that two different-looking expressions are in fact equal. It involves manipulating one side of an equation using known identities until it resembles the other side.

• Start with known identities: Use identities like the Pythagorean identities as anchors to begin transformations.
• Focus on one side: It's useful to concentrate on transforming just one side, usually the more complex one, aiming to make it look like the other.
• Step-by-step simplification: Break down complex fractions, transform functions into basic ones such as sine and cosine, and consolidate terms.

In the given exercise, the left side \( \frac{\tan^2 t - 1}{\sec^2 t} \) is simplified to \( \frac{\tan^2 t - 1}{\tan^2 t + 1} \), a process echoing through identity application and term simplification.
The right side is transformed similarly, reinforcing that both sides are indeed equivalent by arriving at the same considerably simplified expression. Verifying identities not only affirms their correctness but also deepens understanding of trigonometric operations and their interrelationships.