Problem 77 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation \(\frac{\tan ^{2} \gamma+1}{\sec \gamma}=\sec \gamma\) is an identity.

Step 1: Recall Trigonometric Identities

First, let's recall the Pythagorean identity for tangent and secant: \[tan^2 \gamma + 1 = sec^2 \gamma\]This identity will be helpful in transforming the left side of the equation.

Step 2: Rewrite the Left Side of the Equation

Substitute the identity from Step 1 into the left side of the given equation: \[\frac{tan^2 \gamma + 1}{sec \gamma} = \frac{sec^2 \gamma}{sec \gamma}\]

Step 3: Simplify the Expression

Simplify \(\frac{sec^2 \gamma}{sec \gamma}\) by canceling one \(sec \gamma\) from the numerator and the denominator: \[\frac{sec^2 \gamma}{sec \gamma} = sec \gamma\]So, the left side simplifies to the right side of the equation.

Step 4: Verify the Identity

Since both sides of the equation simplify to \(sec \gamma\), the given equation is an identity. Thus, \[\frac{tan^2 \gamma + 1}{sec \gamma} = sec \gamma\] is verified.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental tool in trigonometry. It relates the squares of the sine, cosine, and even tangent functions in a very neat way. The identity states that:

• Per sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• For tangent and secant, it becomes \( \tan^2 \gamma + 1 = \sec^2 \gamma \).

These identities are derived from the Pythagorean Theorem applied in a unit circle. For our exercise, the identity \( \tan^2 \gamma + 1 = \sec^2 \gamma \) was crucial. It allowed us to connect the left side of the equation strongly, transforming it into a more straightforward expression. The Pythagorean identities are not just equations to remember—they serve as bridges that simplify complex trigonometric expressions and help in recognizing pattern solutions.

Simplification

Simplification is the process of breaking down an expression into its simplest form. In the given exercise, simplification was necessary to verify the identity. Here are the steps to simplify complex trigonometric expressions generally:

• Identify common trigonometric identities that can be applied, like the Pythagorean Identity.
• Substitute these identities into your expression.
• Look for opportunities to cancel common terms in fractions.

In the exercise, substituting \( \tan^2 \gamma + 1 \) with \( \sec^2 \gamma \) enabled cancellation of \( \sec \gamma \) in the fraction \( \frac{\sec^2 \gamma}{\sec \gamma} \) making the expression simpler, transforming both sides to be identical, \( \sec \gamma \). Recognizing these opportunities for simplification can make trigonometric problems much easier to solve.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to its side lengths. They are essential in various fields such as physics, engineering, and even finance. The basic trigonometric functions are:

• Sine (\( \sin \))
• Cosine (\( \cos \))
• Tangent (\( \tan \))

Additionally, there are reciprocal functions like secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)). Each function plays a role in evaluating angles and sides in right triangles or wave functions. Understanding how these functions interact with identities, like the Pythagorean Identity, can reveal deeper mathematical relationships. In the context of the exercise, connecting \( \tan^2 \gamma + 1 \) to \( \sec^2 \gamma \) shows how knowing the relationships between these functions allows for the verification and simplification of identities. Learning these interactions strengthens the ability to tackle various trigonometric problems efficiently.