Problem 94 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The equation is verified as an identity.

Step 1: Combine the Left-Hand Side

To verify the identity, start by combining the fractions on the left side of the equation. Add \( \frac{1}{1-\sin \theta} \) and \( \frac{1}{1+\sin \theta} \). The common denominator will be \( (1-\sin \theta)(1+\sin \theta) \). This gives:\[\frac{1 + \sin \theta + 1 - \sin \theta}{(1-\sin \theta)(1+\sin \theta)} = \frac{2}{1 - \sin^2 \theta}\].

Step 2: Use Trigonometric Identity

Recognize that the denominator \(1 - \sin^2 \theta\) can be simplified using the Pythagorean identity. Recall that \(1 - \sin^2 \theta = \cos^2 \theta\). Substitute to get:\[\frac{2}{\cos^2 \theta}\].

Step 3: Simplify to Final Form

Notice that \(\frac{2}{\cos^2 \theta}\) is equivalent to \(2 \sec^2 \theta\) because \( \sec \theta = \frac{1}{\cos \theta} \). Therefore:\[\frac{2}{\cos^2 \theta} = 2 \sec^2 \theta\].

Step 4: Conclude that the Equation is an Identity

Both sides of the original equation equal \(2 \sec^2 \theta\). Thus, the original equation \(\frac{1}{1-\sin \theta} + \frac{1}{1+\sin \theta} = 2 \sec^2 \theta\) is verified as an identity.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It's based on the Pythagorean Theorem and is expressed as:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is crucial because it shows the intrinsic relationship between sine and cosine, derived from the concept of a right triangle.

By rearranging this identity, we derive other forms such as:

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

In our exercise, this identity is used to transform \( 1 - \sin^2 \theta \) to \( \cos^2 \theta \). This step helps simplify the expression immensely, moving us closer to our goal of verifying the trigonometric identity.

Whether you’re dealing with right triangles or unit circles, this identity is a handy tool. It's often used to simplify expressions and solve equations involving trigonometric functions.

Verification of Identities

Verifying identities is a common task in trigonometry and involves proving that two expressions are equal. To verify an identity, like in our exercise, you often start by manipulating one side of the equation to transform it into the other side.

Key strategies for verification include:

• Combining fractions using common denominators
• Applying known identities to simplify expressions
• Expression substitution with simplified forms

In this particular problem, we combined fractions and applied the Pythagorean Identity, realizing that \( 1 - \sin^2 \theta \) is equal to \( \cos^2 \theta \).

Then, by further simplifying the expression, we showed that both sides of the given equation are indeed the same. This confirms the identity, as was achieved through logical step-by-step manipulation.

Trigonometric Functions

Trigonometric functions such as sine (\(\sin\)), cosine (\(\cos\)), and secant (\(\sec\)) are foundational in understanding angles and lengths in right triangles. They extend beyond triangles into waves and circles in calculus.

Each trigonometric function has a reciprocal function, for example:

• Secant (\(\sec\)) is the reciprocal of cosine (\(\cos\)). That means \(\sec \theta = \frac{1}{\cos \theta}\).
• Using these functions in identities helps simplify and manipulate expressions.

In the exercise, the left-hand side was ultimately simplified to match the right-hand side, \(2 \sec^2 \theta\), illustrating the practical use of trigonometric functions in verifying identities.

Understanding these functions and how they relate to one another is crucial for solving a range of mathematical problems. They underpin many principles in both pure and applied mathematics.