Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{\tan ^{2} t+1}{\tan t \csc ^{2} t}=\tan t$$

Step-by-Step Solution

Answer

The equation is verified as an identity.

Step 1: Recall Identity Relationships

The identity \( \tan^2 t + 1 = \sec^2 t \) is one of the Pythagorean identities.

Step 2: Substitute the Pythagorean Identity

Substitute \( \tan^2 t + 1 \) with \( \sec^2 t \) in the left-hand side of the equation: \[ \frac{\sec^2 t}{\tan t \csc^2 t} \]

Step 3: Rewrite Trigonometric Functions

Rewrite \( \sec^2 t \) and \( \csc^2 t \) in terms of sine and cosine:\[ \sec^2 t = \frac{1}{\cos^2 t}, \quad \csc^2 t = \frac{1}{\sin^2 t} \] Substitute these into the equation: \[ \frac{\frac{1}{\cos^2 t}}{\tan t \cdot \frac{1}{\sin^2 t}} \]

Step 4: Simplify the Denominator

We know \( \tan t = \frac{\sin t}{\cos t} \). Thus: \[ \tan t \cdot \frac{1}{\sin^2 t} = \frac{\sin t}{\cos t} \cdot \frac{1}{\sin^2 t} = \frac{1}{\sin t \cos t} \]

Step 5: Simplify the Entire Expression

Substitute and simplify the entire expression:\[ \frac{\frac{1}{\cos^2 t}}{\frac{1}{\sin t \cos t}} = \frac{1}{\cos^2 t} \times \frac{\sin t \cos t}{1} = \frac{\sin t}{\cos t} = \tan t \]The left-hand side simplifies to \( \tan t \), which matches the right-hand side.

Key Concepts

Understanding Pythagorean Identities

Pythagorean identities are fundamental relationships in trigonometry that connect different trigonometric functions. They stem from the Pythagorean theorem, which applies to right-angled triangles. The most common Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). From this base identity, we can derive others, such as:

• \( \tan^2 \theta + 1 = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities are extremely useful for verifying and simplifying trigonometric expressions. In the given exercise, \( \tan^2 t + 1 = \sec^2 t \) represents such a Pythagorean identity. It enables the transformation of more complex trigonometric expressions into simpler ones by substituting one form for another.

Trigonometric Simplification Techniques

Trigonometric simplification involves reducing complex trigonometric expressions into simpler forms. This process uses various identities and algebraic manipulations to make calculations manageable. In the given solution, we simplify the original expression \( \frac{\tan^2 t+1}{\tan t \csc^2 t} \) by first substituting the Pythagorean identity \( \tan^2 t + 1 = \sec^2 t \). This step simplifies the expression to \( \frac{\sec^2 t}{\tan t \csc^2 t} \). The next step uses basic trigonometric conversions, rewriting \( \sec^2 t \) and \( \csc^2 t \) in terms of \( \cos \) and \( \sin \), allowing for further simplifications. We also simplify the denominator by expressing \( \tan t \) in terms of \( \sin t \) and \( \cos t \), leading to a cancellation of terms and ultimately simplifying the expression to match the desired reminder. By understanding how to swap more complex elements for their simpler equivalents, you can effortlessly handle trigonometric expressions. Incorporating identities into simplification will become a powerful ally.

Basic Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are foundational mathematical functions that describe relationships in right-angled triangles and model periodic phenomena like waves. Each function has reciprocal counterparts:

• \( \sin \theta \) (sine) becomes \( \csc \theta \) (cosecant)
• \( \cos \theta \) (cosine) becomes \( \sec \theta \) (secant)
• \( \tan \theta \) (tangent) becomes \( \cot \theta \) (cotangent)

Understanding these functions is critical as they provide the scaffolding for other expressions. For instance, knowing that \( \tan t = \frac{\sin t}{\cos t} \) allows for deeper exploration into how these functions interact through identities and simplifications. Their relationships with each other also enable transformations between different expression forms, as seen in converting \( \sec^2 t \) into \( \frac{1}{\cos^2 t} \) in simplifications. Grasping the fundamental relationships helps you handle diverse problems in trigonometry.