Problem 13 - Precalculus with Calculus Previews

Question

Reduce the given expression to a single trigonometric function. $$ \frac{\sec ^{2} \alpha-1}{\tan \alpha} $$

Step-by-Step Solution

Answer

The expression reduces to \(\tan \alpha\).

Step 1: Identify Trigonometric Identity

Recall the trigonometric identity \[\sec^2 \alpha = 1 + \tan^2 \alpha.\] Using this identity, we will simplify the expression.

Step 2: Substitute the Identity

Replace the expression \(\sec^2 \alpha\) in the numerator with \(1 + \tan^2 \alpha\): \[\frac{\sec^{2} \alpha - 1}{\tan \alpha} = \frac{(1 + \tan^2 \alpha) - 1}{\tan \alpha}.\]

Step 3: Simplify the Numerator

Simplify the numerator:\[\frac{\tan^2 \alpha}{\tan \alpha}.\]

Step 4: Simplify the Entire Expression

Simplify the fraction by canceling \(\tan \alpha\) from the numerator and denominator:\[\frac{\tan^2 \alpha}{\tan \alpha} = \tan \alpha.\] So the expression \(\frac{\sec^{2} \alpha - 1}{\tan \alpha}\) reduces to \(\tan \alpha\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \alpha \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function: \[ \sec \alpha = \frac{1}{\cos \alpha}. \]

• This means the secant function represents how many times the cosine value divides into one.
• It helps in expanding the scope of solving trigonometry problems beyond simple sine and cosine functions.

In trigonometric identities, the secant function is often used alongside its relation to the tangent function and is particularly handy in calculus for integrations and differentiations involving trigonometric terms. By understanding the role of secant, you can also decipher other trigonometric identities, like \( \sec^2 \alpha = 1 + \tan^2 \alpha \), which was pivotal in the original exercise.

Tangent Function

The tangent function, denoted as \( \tan \alpha \), is another essential trigonometric function. It is defined by the ratio of the sine function to the cosine function: \[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha}. \]

• The tangent function represents the slope of the angle \( \alpha \) in the right triangle framework.
• This makes it especially useful for calculations involving gradients and angles in trigonometry.

Typically, the value of \( \tan \alpha \) is infinite when \( \cos \alpha \) is zero; these are points where \( \tan \alpha \) is undefined due to division by zero. The relationship between tangent and secant functions becomes essential when utilizing trigonometric identities like \( \sec^2 \alpha = 1 + \tan^2 \alpha \), allowing one to simplify and solve problems more effectively.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves combining and reducing terms to make calculations more manageable. Understanding identities like \( \sec^2 \alpha = 1 + \tan^2 \alpha \) allows us to replace complex expressions with simpler ones. Here’s a simplified breakdown of the original exercise:- Start by identifying useful trigonometric identities.- Use these identities to substitute expressions.- Simplify iteratively, reducing the expression to a simpler base form.In the given problem:- The expression \( \frac{\sec^{2} \alpha - 1}{\tan \alpha} \) was reduced using the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \).- This substitution turned the original expression into \( \frac{\tan^2 \alpha}{\tan \alpha} \), leading to further cancellation and simplification.- By cancelling the common \( \tan \alpha \) term in the numerator and denominator, the expression reduces to \( \tan \alpha \).This simplification technique is essential because it allows one to solve problems more efficiently and helps deepen the understanding of trigonometric concepts.