Problem 31 - Algebra 2

Question

You can determine whether or not an equation may be a trigonometric identity by graphing the expressions on either side of the equals sign as two separate functions. If the graphs do not match, then the equation is not an identity. If the two graphs do coincide, the equation might be an identity. The equation has to be verified algebraically to ensure that it is an identity. $$ \sec ^{2} x-1=\sin ^{2} x \sec ^{2} x $$

Step-by-Step Solution

Answer

The equation \( \sec^2 x - 1 = \sin^2 x \sec^2 x \) is a trigonometric identity.

Step 1: Understand the Equation

The given equation is \( \sec^2 x - 1 = \sin^2 x \sec^2 x \). We need to test if this is a trigonometric identity. This means the equation should hold true for all values of \( x \) where the functions are defined.

Step 2: Graph the Left Side of the Equation

The left side of the equation is \( \sec^2 x - 1 \). Graph this function using a graphing calculator or software. Observe the pattern and note that \( \sec^2 x - 1 \) is equivalent to \( \tan^2 x \) based on the Pythagorean identity \( \sec^2 x = 1 + \tan^2 x \).

Step 3: Graph the Right Side of the Equation

The right side of the equation is \( \sin^2 x \sec^2 x \). Graph this function separately. Simplify this expression: since \( \sec x = \frac{1}{\cos x} \), \( \sec^2 x = \frac{1}{\cos^2 x} \), so \( \sin^2 x \sec^2 x = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x \).

Step 4: Compare the Graphs

Compare the graphs from steps 2 and 3. Since \( \tan^2 x \) is graphed in both scenarios, the graphs will coincide for the values of \( x \) where both functions are defined. This visual confirmation suggests that the equation might be a trigonometric identity.

Step 5: Algebraically Verify the Identity

To verify algebraically, start by simplifying both sides of the original equation. We already observed that: \( \sec^2 x - 1 = \tan^2 x \) and \( \sin^2 x \sec^2 x = \tan^2 x \). Since both sides simplify to \( \tan^2 x \), the equation \( \sec^2 x - 1 = \sin^2 x \sec^2 x \) holds true for all \( x \) where these functions are defined.

Key Concepts

Graphing Functions

Graphing functions is an essential technique in mathematics to visually understand and verify equations. By plotting two sides of an equation as separate functions, we can immediately see if they coincide, hinting at a potential identity or equality. In the context of trigonometric identities, this involves graphing the expressions on either side of the equation to check if their shapes and positions match.Using a graphing tool, we first graph the left side of the equation, which is given as \( \sec^2 x - 1 \). Upon graphing, it's important to recognize from trigonometric knowledge that \( \sec^2 x - 1 \) simplifies to \( \tan^2 x \), due to the Pythagorean identity.
Next, the right side \( \sin^2 x \sec^2 x \) is graphed. Simplifying this expression algebraically, we find it equals \( \tan^2 x \) as well. Therefore, both sides, when graphed, should produce the same curve, \( \tan^2 x \), within the valid domain of \( x \).The crucial step in graphing functions is not just in plotting, but in comparing these plots. If both sides match for all permissible values of \( x \), we gain a visual cue that the equation might indeed be an identity. However, note that graphing alone is not enough; it serves as an initial check before deeper verification.

Pythagorean Identity

The Pythagorean identity is a crucial component in trigonometry used for simplifying expressions and proving identities. The specific identity used here is \( \sec^2 x = 1 + \tan^2 x \). This identity helps us transform the original equation into an equivalent form, making it easier to compare and verify.When we apply this identity to the left side of the original equation \( \sec^2 x - 1 \), it simplifies directly to \( \tan^2 x \), which is a known derivative of the Pythagorean identity.
This simplification shows the power of these identities in rewriting trigonometric expressions in more recognizable or solvable forms.Understanding and applying the Pythagorean identities efficiently allows us to manipulate and confirm the equivalence of expressions on both sides of the equation. Without such foundational identities, proving the given equation would be much more complex and cumbersome.

Algebraic Verification

Algebraic verification is the process of proving an equation or identity using algebraic manipulation and simplification. This step is essential because even if two graphs coincide, it only suggests rather than confirms the presence of an identity.In this exercise, algebraic verification involves simplifying both sides of the equation independently. We substitute known identities, like using \( \sec^2 x = 1 + \tan^2 x \), to reduce both sides to a common expression, \( \tan^2 x \).
Once simplified, if both sides match (as they do here: \( \sec^2 x - 1 = \tan^2 x \) and \( \sin^2 x \sec^2 x = \tan^2 x \)), the identity is confirmed.One key takeaway is that algebraic verification requires careful handling of trigonometric properties. Identifying and applying correct identities is crucial, and ensures the equality holds universally, for every \( x \) in the domain of the functions involved. This careful approach benefits mathematicians and students alike by ensuring the robustness and reliability of mathematical proofs.