Problem 61 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$(1+\sin \theta)(1-\sin \theta)=\frac{1}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

The identity is verified: \((1+\sin \theta)(1-\sin \theta) = \frac{1}{\sec^2 \theta}.\)

Step 1: Apply the Difference of Squares

Recognize the expression \((1 + \sin \theta)(1 - \sin \theta)\) as a difference of squares. Use the identity \((a + b)(a - b) = a^2 - b^2\). Here, let \(a = 1\) and \(b = \sin \theta\), so we have:\[ (1 + \sin \theta)(1 - \sin \theta) = 1^2 - (\sin \theta)^2 = 1 - \sin^2 \theta. \]

Step 2: Use the Pythagorean Identity

Recall the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Rearrange this identity to express \(1 - \sin^2 \theta\):\[ 1 - \sin^2 \theta = \cos^2 \theta. \]

Step 3: Express Cosine in Terms of Secant

Understand that \(\sec \theta\) is the reciprocal of \(\cos \theta\). Therefore, \(\cos^2 \theta\) can be written as \(\frac{1}{\sec^2 \theta}\), because \(\cos^2 \theta = (\cos \theta)^2 = \left(\frac{1}{\sec \theta}\right)^2 = \frac{1}{\sec^2 \theta}\).

Step 4: Verify the Equality

Substitute the expression for \(\cos^2 \theta\) back into the equation from Step 1. We have:\[ 1 - \sin^2 \theta = \cos^2 \theta = \frac{1}{\sec^2 \theta}. \]Therefore, the identity is verified as \((1+\sin \theta)(1-\sin \theta) = \frac{1}{\sec^2 \theta}.\)

Key Concepts

Difference of Squares

The difference of squares is a foundational algebraic concept that you might have encountered. It deals with expressions in the form

• \((a + b)(a - b) = a^2 - b^2\)

Here,

• \(a\) and \(b\) can be any expressions, not just simple numbers.

When you apply this identity, the result is a subtraction of the square of \(b\) from the square of \(a\). This property helps simplify expressions like \((1 + \sin \theta)(1 - \sin \theta)\).
In this example, you let \(a = 1\) and \(b = \sin \theta\), which leads to

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

Recognizing and applying the difference of squares allows you to move forward in solving complex trigonometric identities by simplifying them to more manageable forms.

Pythagorean Identity

The Pythagorean identity is one of the most critical trigonometric identities. It links sine and cosine functions in a very neat relationship. The identity states:

• \(\sin^2 \theta + \cos^2 \theta = 1\).

It's named after the famous Pythagorean Theorem due to its connection to right-angled triangles. When you need to verify identities or solve equations, rearranging this equation can be very useful.
In the given problem, using this identity allows you to replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\). This transformation is pivotal because it helps convert the expression into a form where other identities, like reciprocal identities, can be applied.

Secant Function

The secant function, denoted by \(\sec \theta\), is a less commonly used trigonometric function that represents the reciprocal of the cosine function.

• Formally, \(\sec \theta = \frac{1}{\cos \theta}\).

When solving trigonometric identities, understanding how secant interacts with cosine is crucial.

• This relationship implies \((\cos \theta)^2 = \left(\frac{1}{\sec \theta}\right)^2\) or \(\cos^2 \theta = \frac{1}{\sec^2 \theta}\).

In the context of the exercise, substituting \(\cos^2 \theta\) with \(\frac{1}{\sec^2 \theta}\) is a key step. This shift makes verifying the identity possible, as you match the left side of the equation to the right side: \(\frac{1}{\sec^2 \theta}\).

Reciprocal Identities

Reciprocal identities play a big role in trigonometry by connecting base trigonometric functions to their reciprocals. These identities allow you to transform expressions using their reciprocal counterparts:

• For cosine, the reciprocal identity is \(\sec \theta = \frac{1}{\cos \theta}\).
• Similarly, for sine and tangent functions, their reciprocals are cosecant and cotangent, respectively.

For this exercise, specifically, understanding the cosine-secant reciprocal identity is essential. You need to recognize how these identities allow conversion between different trigonometric functions, making it possible to simplify complex expressions.
Once you rewrite \(\cos^2 \theta\) in terms of \(\sec^2 \theta\), it confirms the identity \(1 - \sin^2 \theta = \cos^2 \theta = \frac{1}{\sec^2 \theta}\), completing the verification.