Problem 15 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{\sec ^{2} x-1}{\sec ^{2} x} $$

Step-by-Step Solution

Answer

The simplified expression is \( \tan^2 x / \sec^2 x = \sin^2 x \).

Step 1: Recall a Trigonometric Identity

We start by identifying a trigonometric identity that relates to the given expression. We know that \( \sec^2 x = 1 + \tan^2 x \).

Step 2: Substitute the Identity

We substitute the identity \( \sec^2 x = 1 + \tan^2 x \) into the expression:\[\frac{\sec^2 x - 1}{\sec^2 x} = \frac{(1+\tan^2 x) - 1}{1+\tan^2 x}.\]

Step 3: Simplify the Numerator

Simplify the expression in the numerator:\[(1 + \tan^2 x) - 1 = \tan^2 x.\]

Step 4: Formulate the Simplified Expression

Replace the simplified numerator back in the expression:\[\frac{\tan^2 x}{1+\tan^2 x}.\]

Step 5: Simplify the Fraction

The expression \( \frac{\tan^2 x}{1 + \tan^2 x} \) is already in its simplest form, as \( \tan^2 x \) cannot be further reduced with the denominator \( 1 + \tan^2 x \). Thus, \( \frac{\tan^2 x}{\sec^2 x} = \sin^2 x \).

Key Concepts

Secant Function

The secant function, written as \( \sec x \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the cosine function. So, \( \sec x = \frac{1}{\cos x} \). This means that for any angle \( x \), the secant value is simply one divided by the cosine of that angle.

Here are some essential points to remember about the secant function:

• It has the same sign as the cosine function because dividing by a positive number won't change the sign.
• \( \sec x \) is undefined wherever \( \cos x = 0 \), because division by zero is undefined.
• Its range is \( (-\infty, -1] \cup [1, \infty) \).
• In a right triangle setting, for acute angles, \( \sec x \) represents the ratio of the hypotenuse over the adjacent side.

Due its reciprocal nature, the secant function is used mainly in trigonometric identities and solving equations, as seen in this exercise. The identity \( \sec^2 x = 1 + \tan^2 x \) connects the secant function to the tangent function.

Tangent Function

The tangent function, typically noted as \( \tan x \), is another fundamental trigonometric function. It describes the relationship between the sine and cosine functions, being the quotient of sine over cosine: \( \tan x = \frac{\sin x}{\cos x} \).

A few critical features of the tangent function include:

• It is undefined at angles where the cosine value is zero (such as \( x = 90^\circ, 270^\circ \), etc.), leading to vertical asymptotes in its graph.
• The tangent function repeats every \( 180^\circ \) or \( \pi \) radians, making it a periodic function.
• Its range is all real numbers \( (-\infty, \infty) \).
• In the context of a right triangle, \( \tan x \) represents the ratio of the opposite side to the adjacent side.

The identity used in our solution, \( \sec^2 x = 1 + \tan^2 x \), is a direct result of rearranging the Pythagorean identity related to sine and cosine and is pivotal for simplifying expressions that involve secants squared.

Trigonometric Simplification

Trigonometric simplification involves rewriting trigonometric expressions into more manageable or elegant forms, often employing various identities. These identities are relationships among trigonometric functions that are true for all valid angles.

In this exercise, we demonstrated simplification using a known identity:

• The identity \( \sec^2 x = 1 + \tan^2 x \) was used to transform the complex expression into a simpler form by substitution.
• By substituting \( \sec^2 x \) with \( 1 + \tan^2 x \), the expression became more straightforward, as shown: \[ \frac{(1 + \tan^2 x) - 1}{1 + \tan^2 x} = \frac{\tan^2 x}{1 + \tan^2 x} \]
• Simplification often breaks down complex expressions like this into sums, differences, or fractions that are easier to understand or further manipulate.

In particular, trigonometric simplification is instrumental when dealing with equations or expressions required in calculus, physics, and engineering fields, as it simplifies the calculation and understanding of complex trigonometric forms. Mastering these identities and simplification techniques can greatly enhance problem-solving efficiency.