Problem 71 - Algebra and Trigonometry

Question

Verify the identity. $$ \sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x $$

Step-by-Step Solution

Answer

The identity is verified by using trigonometric identities and algebraic simplifications.

Step 1: Rewrite Using Basic Identities

Recall the basic trigonometric identities: \( \sec^2 x = 1 + \tan^2 x \). This identity will help us rewrite components of the given expression.

Step 2: Expand Left Side

We start with the left side: \( \sec^4 x - \tan^4 x \). Use the formula \( a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) \). Thus: \( \sec^4 x - \tan^4 x = (\sec^2 x + \tan^2 x)(\sec^2 x - \tan^2 x) \).

Step 3: Substitute Identity

Now substitute \( \sec^2 x - \tan^2 x \) using the identity \( \sec^2 x = 1 + \tan^2 x \). So \( \sec^2 x - \tan^2 x = (1 + \tan^2 x) - \tan^2 x = 1 \).

Step 4: Simplify and Verify

Replace \( \sec^2 x - \tan^2 x \) with \( 1 \) in our factorization to get: \( (\sec^2 x + \tan^2 x) \times 1 = \sec^2 x + \tan^2 x \). This matches the right side, verifying the identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is a reciprocal trigonometric function. It is defined as the reciprocal of the cosine function:

• \( \sec x = \frac{1}{\cos x} \)

The secant function is particularly important when dealing with identities and equations related to trigonometry.
In many trigonometric identities, secant often appears in the form of \( \sec^2 x \) because of its relationship with the tangent function.
One of the key properties of the secant function is that since \( \cos x = 0 \) where \( x = (2n+1)\frac{\pi}{2} \) for any integer \( n \), \( \sec x \) will be undefined at these points.
In terms of symmetry, the graph of \( \sec x \) resembles an upside-down bell curve, reflecting over the line \( y = 1 \) and extending vertically towards infinity as it approaches its undefined points.

Tangent Function

The tangent function, represented by \( \tan x \), is one of the basic trigonometric functions, derived from the ratio of the sine and cosine functions:

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

This function is particularly powerful in describing angles and triangles in trigonometry.
The tangent function repeats every \( \pi \) radians, leading to a periodic graph with specific points where it's undefined (where \( \cos x = 0 \), such as \( x = (2n+1)\frac{\pi}{2} \) for any integer \( n \)).
Additionally, tangent has unique identities such as \( \sec^2 x = 1 + \tan^2 x \), connecting deeply with the secant function.
This identity was crucial in the solution process of verifying the given trigonometric identity. Understanding \( \tan^2 x \) and \( \sec^2 x \) simplifies manipulation in various trigonometric equations and identities.

Trigonometric Equations

Trigonometric equations involve expressions containing trigonometric functions that we solve for an unknown variable, often an angle.
These equations can be used to verify identities or solve for exact angle measures.
They are intricate because they require a deep understanding of the properties of trigonometric functions and their relationships, such as secant and tangent.

Common Techniques

• Factorization: Splitting an equation into simpler components to find solutions.
• Using Identities: Leveraging known identities (like \( \sec^2 x = 1 + \tan^2 x \)) to simplify and solve expressions.
• Substitution: Replacing a trigonometric function using its equivalency (e.g., \( \sec^2 x = 1 + \tan^2 x \)).

Verification of Identities

Verifying identities, as done in the problem, turns into systematic manipulation:

• Changing forms: Connecting terms through identities.
• Stepwise simplification: Continually using algebraic restructuring and trigonometric definitions.

Grasping these concepts not only aids in solving exercises but develops a comprehensive understanding of how trigonometric functions interact.