Problem 32 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{1+\sin t}{\cos t}+\frac{\cos t}{1+\sin t}=2 \sec t $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(2 \sec t\).

Step 1: Identify the Given Identity

The identity to verify is: \[\frac{1+\sin t}{\cos t}+\frac{\cos t}{1+\sin t}=2 \sec t\]

Step 2: Find a Common Denominator

To combine \(\frac{1+\sin t}{\cos t}+\frac{\cos t}{1+\sin t}\), let's find a common denominator. The least common denominator (LCD) for these two fractions is \(\cos t (1 + \sin t)\).

Step 3: Rewrite Each Fraction with the Common Denominator

Convert each fraction:- \(\frac{1+\sin t}{\cos t} \rightarrow \frac{(1+\sin t)(1+\sin t)}{\cos t(1+\sin t)}\)- \(\frac{\cos t}{1+\sin t} \rightarrow \frac{\cos t \cos t}{\cos t(1+\sin t)}\)Thus, the expression becomes: \[\frac{(1+\sin t)^2 + \cos^2 t}{\cos t (1 + \sin t)}\]

Step 4: Simplify the Numerator

Expand the numerator:\[(1+\sin t)^2 = 1 + 2\sin t + \sin^2 t\]So:\[(1+\sin t)^2 + \cos^2 t = 1 + 2\sin t + \sin^2 t + \cos^2 t\]Since \(\sin^2 t + \cos^2 t = 1\), it simplifies to:\[1 + 2\sin t + 1 = 2 + 2\sin t\]

Step 5: Simplify the Entire Expression

Substitute the simplified numerator back into the fraction:\[\frac{2 + 2\sin t}{\cos t(1 + \sin t)} = \frac{2(1 + \sin t)}{\cos t(1 + \sin t)}\]Cancelling \(1 + \sin t\) from the numerator and the denominator gives:\[\frac{2}{\cos t}\]

Step 6: Recognize the Result as the Trigonometric Identity

The simplified expression \(\frac{2}{\cos t}\) is the same as \(2 \sec t\), since \(\sec t = \frac{1}{\cos t}\). Thus, the identity is verified.

Key Concepts

Verifying Trigonometric Identities

Verifying trigonometric identities involves demonstrating that two sides of an equation are equal by using known trigonometric identities and algebraic manipulations. This exercise requires verifying the identity \(\frac{1+\sin t}{\cos t}+\frac{\cos t}{1+\sin t}=2 \sec t\). To prove such identities, it's crucial to manipulate one side of the equation to transform it into the other, rather than assuming the equation is true from the start.Here are some tips to streamline the verification process:

• Start by identifying a strategy, such as combining fractions or substituting common identities.
• Focus on transforming the more complex side of the equation or the side that suggests a straightforward identity transformation.
• Remember frequently used identities, such as \(\sin^2 t + \cos^2 t = 1\), to simplify the process.

Understanding these principles can help you not only in solving this problem but also in handling similar exercises.

Common Denominator in Fractions

Finding a common denominator is a crucial algebraic step when adding or subtracting fractions. In this problem, to verify the identity \(\frac{1+\sin t}{\cos t}+\frac{\cos t}{1+\sin t}\), finding the least common denominator (LCD) is key to combining these fractions.To identify the common denominator:

• Examine each fraction's denominator: the first fraction has \(\cos t\) and the second \(1+\sin t\).
• The LCD here is \(\cos t(1+\sin t)\).

Once the LCD is established, rewrite each fraction to have this common denominator:

• The first fraction becomes: \(\frac{(1+\sin t)(1+\sin t)}{\cos t(1+\sin t)}\).
• The second fraction becomes: \(\frac{\cos t \cos t}{\cos t(1+\sin t)}\).

Now the fractions have a common basis to be combined, enabling further simplification.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is about applying identities and algebraic laws to cleanly and accurately transform an expression. After combining fractions in this exercise, the expression was simplified effectively using the Pythagorean identity.Let's simplify the combined expression \(\frac{(1+\sin t)^2 + \cos^2 t}{\cos t (1 + \sin t)}\):

• Begin by expanding \((1+\sin t)^2 = 1 + 2\sin t + \sin^2 t\).
• Add \(1 + 2\sin t + \sin^2 t\) to \(\cos^2 t\).

Using the Pythagorean identity, \(\sin^2 t + \cos^2 t = 1\), allows you to simplify the numerator:

• Replace \(\sin^2 t + \cos^2 t\) with 1.
• The numerator becomes \(2 + 2\sin t\).

This can then be rewritten as \(\frac{2(1 + \sin t)}{\cos t(1 + \sin t)}\). Upon canceling \(1 + \sin t\) from both the numerator and denominator, the resulting expression is \(\frac{2}{\cos t}\). Recognizing that this is equivalent to \(2 \sec t\) successfully verifies the given trigonometric identity.