Problem 33 - Contemporary Precalculus

Question

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(1-\sec t)^{2}$$

Step-by-Step Solution

Answer

Question: Expand and simplify the expression \((1-\sec{t})^2\). Answer: The simplified expression is \(\frac{1-2\cos{t}+\sin^2{t}}{\cos^2{t}}\).

Step 1: Rewrite the expression using secant identity

Firstly, we need to rewrite the expression, \((1-\sec{t})^2\), using the secant identity. So, replace \(\sec{t}\) with \(\frac{1}{\cos{t}}\). The expression becomes: $$(1-\frac{1}{\cos{t}})^2$$

Step 2: Calculate the square of the expression

Now, we will expand the expression and square it: $$\left(1-\frac{1}{\cos{t}}\right)^2=\left(\frac{\cos{t}-1}{\cos{t}}\right)^2=\frac{(\cos{t}-1)^2}{\cos^2{t}}$$

Step 3: Expand the numerator

Expand the numerator by multiplying \((\cos{t}-1)\) by itself: $$\frac{(\cos{t}-1)^2}{\cos^2{t}}=\frac{\cos^2{t}-2\cos{t}+1}{\cos^2{t}}$$

Step 4: Simplify the expression using trigonometric identity

Use the Pythagorean identity, \(\sin^2{t}+\cos^2{t}=1\), to simplify the expression further. Since we have \(\cos^2{t}\) in the numerator, we can rewrite this identity as \(\sin^2{t}=1-\cos^2{t}\). Now, substitute \(\sin^2{t}=1-\cos^2{t}\) in the expression: $$\frac{\cos^2{-}2\cos{t}+1}{\cos^2{t}}=\frac{(1-\sin^2{t}){-}2\cos{t}+1}{\cos^2{t}}=\frac{1-2\cos{t}+\sin^2{t}}{\cos^2{t}}$$ So the final simplified expression is: $$\frac{1-2\cos{t}+\sin^2{t}}{\cos^2{t}}$$

Key Concepts

Secant Identity

Understanding the secant identity is crucial for simplifying trigonometric expressions. The secant of an angle, represented as \( \text{sec}(t) \), is defined as the reciprocal of the cosine of that angle. In essence, \( \text{sec}(t) = \frac{1}{\text{cos}(t)} \). This definition is pivotal in transforming complex trigonometric expressions into more manageable forms.

When faced with an expression involving the secant function, such as \( (1-\text{sec}(t))^2 \), the first step is to substitute \( \text{sec}(t) \) with \( \frac{1}{\text{cos}(t)} \). Rewriting functions in terms of sine and cosine can often lead to further simplifications, as these are the fundamental trigonometric functions upon which most identities are built.

Trigonometric Simplification

The process of trigonometric simplification involves rewriting trigonometric expressions in a simpler or more compact form. This often involves applying various trigonometric identities and algebraic manipulations. To simplify an expression like \( \frac{(\text{cos}(t) - 1)^2}{\text{cos}^2(t)} \), we expand the numerator and look for opportunities to simplify the expression.

For instance, after expansion, you might have terms that are common in both the numerator and the denominator that can be canceled out. Another approach is to use known identities to transform parts of the expression into equivalent forms that are easier to work with or combine. Understanding these strategies is essential for efficiency and accuracy in solving trigonometry problems.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, expressing a fundamental relationship between the sine and cosine of an angle. The identity states that \( \text{sin}^2(t) + \text{cos}^2(t) = 1 \), which can be rearranged depending on the needs of the problem at hand. For example, we can isolate \( \text{sin}^2(t) \) to get \( \text{sin}^2(t) = 1 - \text{cos}^2(t) \).

This powerful identity allows us to convert sine-squared terms to cosine-squared terms or vice versa, significantly simplifying expressions where both sine and cosine functions appear. Referring back to the original exercise, this identity is applied to replace \( \text{cos}^2(t) \) with \( 1 - \text{sin}^2(t) \) in the expression, helping to achieve the final simplified result.