Problem 28 - Precalculus Mathematics for Calculus

Question

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$\frac{\tan y}{\csc y}=\sec y-\cos y$$

Step-by-Step Solution

Answer

The equation is verified as an identity both algebraically and graphically.

Step 1: Simplify the Left Side

To verify that the equation is an identity, start by simplifying the left-hand side (LHS) of the equation. The LHS is \( \frac{\tan y}{\csc y} \). Recall the trigonometric identities, \( \tan y = \frac{\sin y}{\cos y} \) and \( \csc y = \frac{1}{\sin y} \). Substitute these into the LHS: \[ \frac{\tan y}{\csc y} = \frac{\frac{\sin y}{\cos y}}{\frac{1}{\sin y}} = \frac{\sin^2 y}{\cos y}. \]

Step 2: Simplify the Right Side

Now simplify the right-hand side (RHS) of the equation. The RHS is \( \sec y - \cos y \). Substitute the identity for \( \sec y \), which is \( \sec y = \frac{1}{\cos y} \): \[ \sec y - \cos y = \frac{1}{\cos y} - \cos y. \] To subtract fractions, express \( \cos y \) as \( \frac{\cos y}{1} \) and find a common denominator: \[ \frac{1}{\cos y} - \frac{\cos^2 y}{\cos y} = \frac{1-\cos^2 y}{\cos y}. \]

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 y + \cos^2 y = 1 \). Substitute \( 1-\cos^2 y \) with \( \sin^2 y \) in the RHS: \[ \frac{1-\cos^2 y}{\cos y} = \frac{\sin^2 y}{\cos y}. \] Now, both the LHS and RHS are \( \frac{\sin^2 y}{\cos y} \), showing that the original equation is an identity.

Step 4: Confirm Graphically

To confirm graphically, plot both \( \frac{\tan y}{\csc y} \) and \( \sec y - \cos y \) on a graphing calculator or software. Ensure you set the domain to avoid undefined regions, typically where 'y' is a multiple of \( \frac{\pi}{2} \) for trigonometric functions. Check that the graphs coincide entirely, illustrating that the expression is true for all values within the domain. This visual confirmation supports the algebraic proof.

Key Concepts

Algebraic Proofs

When verifying a trigonometric identity using algebraic proofs, our main goal is to show that both sides of the given equation are equal. This involves simplifying each side of the equation using trigonometric identities. For example, let's consider the equation

• \( \frac{\tan y}{\csc y} = \sec y - \cos y \)

To begin the verification process, we simplify the left-hand side (LHS). By recalling that \( \tan y = \frac{\sin y}{\cos y} \) and \( \csc y = \frac{1}{\sin y} \), we substitute these into the LHS to get:

• \( \frac{\tan y}{\csc y} = \frac{\sin^2 y}{\cos y} \)

Next, we simplify the right-hand side (RHS) by substituting \( \sec y = \frac{1}{\cos y} \), leading to:

• \( \sec y - \cos y = \frac{1 - \cos^2 y}{\cos y} \)

Using the Pythagorean identity, \( \sin^2 y + \cos^2 y = 1 \), we replace \( 1 - \cos^2 y \) with \( \sin^2 y \), resulting in the RHS becoming \( \frac{\sin^2 y}{\cos y} \).
Since both sides reduce to \( \frac{\sin^2 y}{\cos y} \), we have proven the identity algebraically.

Graphical Verification

Graphical verification of trigonometric identities involves plotting both sides of the equation on a graph to observe if they coincide. This method offers a visual confirmation of the algebraic proof.
In our example, we need to plot \( \frac{\tan y}{\csc y} \) and \( \sec y - \cos y \) on the same graph using a graphing tool. Before plotting, it's vital to set the correct domain to avoid points where the functions are undefined, commonly occurring at multiples of \( \frac{\pi}{2} \).
When plotted, observe whether the graphs completely overlap for the valid domain. If they do, this visual evidence supports the fact that the two sides of the equation are indeed identical for every value within the domain. This verification complements our algebraic solution, making the understanding of the identity more concrete and robust.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a crucial skill in verifying identities. It involves using known identities to transform complex expressions into simpler, equivalent forms. This not only aids in recognizing identities but also in solving trigonometric equations effectively.
In practice, simplify expressions by:

• Substituting basic trigonometric identities such as \( \tan y = \frac{\sin y}{\cos y} \) and \( \csc y = \frac{1}{\sin y} \).
• Looking for opportunities to apply fundamental identities like the Pythagorean identity, \( \sin^2 y + \cos^2 y = 1 \).
• Rewriting expressions in terms of a single function when possible to achieve a simplified form.

Through this approach, seemingly complex trigonometric equations can typically be reduced to easier-to-handle forms, aiding in both proving identities and further calculations. A deep understanding of these methods simplifies both the learning and application of trigonometry in various mathematical problems.