Problem 24 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\sec ^{4} u-\sec ^{2} u=\tan ^{2} u+\tan ^{4} u$$

Step-by-Step Solution

Answer

The identity is verified: \( \sec^{4}u - \sec^{2}u = \tan^{2}u + \tan^{4}u \).

Step 1: Recall Trigonometric Identities

To verify the identity \( \sec^{4}u - \sec^{2}u = \tan^{2}u + \tan^{4}u \), start by recalling that \( \sec^{2}u = 1 + \tan^{2}u \). We will use this identity to manipulate the given expression.

Step 2: Rewrite the Left Side

Rewrite the left side using the identity \( \sec^{2}u = 1 + \tan^{2}u \):\[ \sec^{4}u = (\sec^{2}u)^2 = (1 + \tan^{2}u)^2. \] This expands to \[ \sec^{4}u = 1 + 2\tan^{2}u + \tan^{4}u. \] Now subtract \( \sec^{2}u \) from this expression: \[ \sec^{4}u - \sec^{2}u = (1 + 2\tan^{2}u + \tan^{4}u) - (1 + \tan^{2}u). \]

Step 3: Simplify the Left Side

Combine like terms after subtraction: \[ 1 + 2\tan^{2}u + \tan^{4}u - 1 - \tan^{2}u = \tan^{2}u + \tan^{4}u. \] This simplifies to \( \tan^{2}u + \tan^{4}u \), which matches the right side of the original equation.

Step 4: Verify the Identity

Both sides of the original equation now match: \( \sec^{4}u - \sec^{2}u = \tan^{2}u + \tan^{4}u \). We have successfully verified the identity.

Key Concepts

Secant Function

The secant function is essentially the reciprocal of the cosine function. In trigonometry, it is used to relate the angles and sides of a right-angled triangle, and it can be represented as:

• \( \sec(u) = \frac{1}{\cos(u)} \)

This makes the secant function vital in various trigonometric identities. In our example, the given identity includes \( \sec^{4}u \) and \( \sec^{2}u \). To evaluate expressions involving secant, we often rely on fundamental identities like \( \sec^{2}u = 1 + \tan^{2}u \). These identities serve as tools to simplify and transform the expressions correctly for verification or further calculations.
Thus, understanding the secant function and its properties is crucial in breaking down and solving trigonometric identities effectively.

Tangent Function

The tangent function is another fundamental function in trigonometry. It connects the sine and cosine functions and is defined as:

• \( \tan(u) = \frac{\sin(u)}{\cos(u)} \)

The tangent function helps in expressing angles and understanding trigonometric relationships. In our exercise solution, it plays a central role through the identity \( \sec^{2}u = 1 + \tan^{2}u \), which allows us to solve and verify the equation.
The exercise involves expressions like \( \tan^{2}u \) and \( \tan^{4}u \), which showcase the form and empowerment of trigonometric functions in simplifying complex expressions. Understanding the properties of the tangent function and how it relates to secant and other trigonometric functions is crucial for tackling such identities.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two expressions are equivalent through algebraic manipulation and the application of trigonometric identities. The purpose is to transform and simplify the equations until both sides match, as seen in our example.
Here are steps often used in the process:

• Rewrite one or both sides using known identities, like \( \sec^{2}u = 1 + \tan^{2}u \).
• Apply algebraic operations, such as expanding and simplifying terms.
• Seek to make both sides look identical, confirming the original equation's truth.

When verifying identities, the goal is to strategically use mathematical tools to ensure clarity and correctness. It involves looking at the structure, decomposing complex parts, and finding a common form. Mastering this process enhances problem-solving skills in trigonometry.