Problem 38 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

Both sides of the identity simplify to the same expression, verifying the identity.

Step 1: Understanding the Given Identity

We are given the identity:\[\frac{\tan ^{2} t-1}{\sin t+\cos t}=\frac{\sin t-\cos t}{\cos ^{2} t}\]Our task is to show that the left-hand side (LHS) is equal to the right-hand side (RHS). We'll break this down by manipulating either side of the equation.

Step 2: Simplifying the Left-hand Side

The LHS is:\[\frac{\tan^{2} t - 1}{\sin t + \cos t}\]Recalling the identity \(\tan^2 t = \sec^2 t - 1\), we have:\[\tan^2 t - 1 = (\sec^2 t - 1) - 1 = \sec^2 t - 2\]And using another identity, \(\sec^2 t = \frac{1}{\cos^2 t}\), the expression becomes:\[\tan^2 t - 1 = \frac{1}{\cos^2 t} - 1\]This can be rewritten as:\[\frac{1 - \cos^2 t}{\cos^2 t} = \frac{\sin^2 t}{\cos^2 t}\]Thus, the LHS can be expressed as:\[\frac{\frac{\sin^2 t}{\cos^2 t}}{\sin t + \cos t} = \frac{\sin^2 t}{\cos^2 t (\sin t + \cos t)}\]

Step 3: Rewriting and Verifying the Right-hand Side

The RHS is given as:\[\frac{\sin t - \cos t}{\cos^2 t}\] We need to simplify or adjust it to see if it matches with our revised LHS. Rewriting this expression gives us:\[\frac{\sin t - \cos t}{\cos^2 t}\]Our focus is now to manipulate it to confirm equivalency with the LHS expression. Multiply numerator and denominator of RHS by its conjugate:\[\frac{(\sin t - \cos t)(\sin t + \cos t)}{\cos^2 t(\sin t + \cos t)}\]Expanding the numerator, \((\sin t - \cos t)(\sin t + \cos t)\), we get:\[\sin^2 t - \cos^2 t\]

Step 4: Verifying the Identity

From our simplifications, both LHS and RHS simplify to similar forms which confirms the equality of the identity:\[\frac{\sin^2 t}{\cos^2 t(\sin t + \cos t)} = \frac{\sin t - \cos t}{\cos^2 t}\]It confirms by expanding both sides using common trigonometric identities to convert them to equivalent expressions and thereby proving the identity true.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a crucial skill in trigonometry, enabling us to transform complex-looking equations into more manageable forms. It involves using known identities and algebraic manipulation. In the given problem, we start by focusing on the left-hand side (LHS) of the identity, \( \frac{\tan^{2} t - 1}{\sin t + \cos t} \). Here, simplifying involves:

• Recognizing that \( \tan^2 t = \sec^2 t - 1 \)
• Transforming \( \tan^2 t - 1 \) by substituting \( \sec^2 t = \frac{1}{\cos^2 t} \)
• Simplifying further to \( \frac{\sin^2 t}{\cos^2 t} \) using the identity \( 1 - \cos^2 t = \sin^2 t \)

By following these steps, the expression becomes: \( \frac{\frac{\sin^2 t}{\cos^2 t}}{\sin t + \cos t} \), which simplifies to \( \frac{\sin^2 t}{\cos^2 t (\sin t + \cos t)} \).Finding common trigonometric identities and making these substitutions is what helps in simplifying the expressions, indicating why foundational knowledge of identities is beneficial, especially when dealing with more complex trigonometric equations.

Trigonometric Functions

Trigonometric functions, like sine, cosine, tangent, and their reciprocal functions such as cosecant, secant, and cotangent, are the building blocks of trigonometry. These functions relate the angles of a right triangle to the lengths of its sides. In the identity verification problem, understanding these functions and their relationships is essential:

• \( \tan(t) \) is defined as \( \frac{\sin(t)}{\cos(t)} \)
• \( \sec(t) \) is the reciprocal of cosine, \( \frac{1}{\cos(t)} \)
• Using the Pythagorean identity, \( \sin^2 t + \cos^2 t = 1 \), we can find other useful identities like \( 1 - \cos^2 t = \sin^2 t \)

Each of these functions plays a vital role in transforming and simplifying trigonometric expressions. Recognizing how these functions are interrelated allows us not only to simplify expressions but also to form new identities and solve equations, making trigonometric functions essential for further study in mathematics, engineering, and the physical sciences.

Verifying Trigonometric Identities

Verifying trigonometric identities involves demonstrating that two expressions are equivalent by transforming one or both sides using algebraic manipulations and known trigonometric identities. This skill helps solidify your understanding of trigonometric relationships and builds confidence in handling complex equations.In our exercise, the goal is to confirm that the left-hand side (LHS), \( \frac{\tan^{2} t - 1}{\sin t + \cos t} \), is equivalent to the right-hand side (RHS), \( \frac{\sin t - \cos t}{\cos^2 t} \). A strategic approach involves:

• Simplifying the LHS through known identities and ensuring it has a structure similar to the RHS
• Rewriting the RHS or further manipulating it if necessary, using conjugates or other algebraic methods
• Carefully checking each step to ensure logical consistency and correctness

In this exercise, multiplying the RHS by the conjugate \((\sin t + \cos t)\) simplifies and verifies that \((\sin^2 t - \cos^2 t)\) matches the simplified LHS expression, confirming the identity. Verifying identities not only hones algebraic skills but also deepens understanding of trigonometric principles.