Problem 7 - Precalculus Mathematics for Calculus

Question

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$\tan ^{2} x-\sec ^{2} x$$

Step-by-Step Solution

Answer

The expression simplifies to \(-1\).

Step 1: Express in Terms of Sine and Cosine

Start by expressing \( \tan x \) and \( \sec x \) in terms of sine and cosine:\[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \sec x = \frac{1}{\cos x} \]These identities will help us rewrite the given expression.

Step 2: Substitute and Expand

Substitute the identities for \( \tan x \) and \( \sec x \) into the expression \( \tan ^{2} x-\sec ^{2} x \):\[\left( \frac{\sin x}{\cos x} \right)^2 - \left( \frac{1}{\cos x} \right)^2 \]Simplify each term:\[\frac{\sin^2 x}{\cos^2 x} - \frac{1}{\cos^2 x} \]

Step 3: Combine Fractions

Since both terms have the same denominator, combine them:\[\frac{\sin^2 x - 1}{\cos^2 x} \]

Step 4: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Rearrange it to express \( 1 - \sin^2 x \) as \( \cos^2 x \):\[ 1 - \sin^2 x = \cos^2 x \]Use this to rewrite \( \sin^2 x - 1 \) as:\[ -(1 - \sin^2 x) = -\cos^2 x \]

Step 5: Simplify the Expression

Substitute \( -\cos^2 x \) into the combined fraction:\[\frac{-\cos^2 x}{\cos^2 x} \]Simplify it to:\[ -1 \]The expression simplifies to \( -1 \).

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry, often abbreviated as "sin." It is a periodic function that represents the y-coordinate position on the unit circle. The function is defined for an angle \( x \) and is written as \( \sin x \). This function helps in understanding various trigonometric identities and is essential in expressing other trigonometric functions such as tangent. The sine function oscillates between -1 and 1, which means it never exceeds these values.

• Defined as \( \sin x = \frac{\text{opposite side}}{\text{hypotenuse}} \) in a right triangle.
• Periodicity: \( \sin(x + 2\pi) = \sin x \).
• Symmetry: \( \sin(-x) = -\sin x \).

Understanding these properties of the sine function helps in simplifying expressions in terms of sine and cosine. It's especially useful in problems involving identities and transformations.

Cosine Function

The cosine function, denoted as "cos," is another crucial trigonometric function. It measures the x-coordinate on the unit circle for a given angle \( x \), complementing the sine function by defining its relationship through the right triangle. The cosine function is equally periodic and its values range from -1 to 1, similar to the sine function. It is expressed as \( \cos x \) and is often used alongside the sine function to express other trigonometric identities.

• Defined as \( \cos x = \frac{\text{adjacent side}}{\text{hypotenuse}} \) in a right triangle.
• Periodicity: \( \cos(x + 2\pi) = \cos x \).
• Symmetry: \( \cos(-x) = \cos x \).

In trigonometric transformations and simplifications, the cosine function works hand-in-hand with the sine function. Understanding its relationship with the sine function through identities can simplify and reduce complex trigonometric expressions.

Pythagorean Identity

One of the most important and widely used identities in trigonometry is the Pythagorean identity. It is expressed as \( \sin^2 x + \cos^2 x = 1 \). This identity forms the foundation for many mathematical simplifications, as it establishes the fundamental relationship between the square of the sine and cosine functions. This identity can be rearranged to solve for any one of the functions in terms of the other, such as \( \cos^2 x = 1 - \sin^2 x \) or \( \sin^2 x = 1 - \cos^2 x \).

• Derived from the Pythagorean theorem in geometry.
• Helps in simplifying expressions by converting square terms.
• Assists in converting different trigonometric forms within identities.

In problems like the one presented in the exercise, this identity allows us to convert expressions and reach simplifications like reducing a complex expression to \(-1\). Understanding and using the Pythagorean identity is fundamental in mastering trigonometric transformations and simplifications.