Problem 52 - Algebra 2

Question

Simplify each expression. 4\(\left(\sec ^{2} \theta-\frac{\sin ^{2} \theta}{\cos ^{2} \theta}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to 4.

Step 1: Rewrite Using Trigonometric Identities

We begin by recognizing that \(\frac{\sin^2 \theta}{\cos^2 \theta}\) is equal to \(\tan^2 \theta\) because \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). So, we rewrite the expression as:\[4 \left( \sec^2 \theta - \tan^2 \theta \right)\]

Step 2: Use the Pythagorean Identity

We use the Pythagorean identity \(\sec^2 \theta - \tan^2 \theta = 1\). This simplifies the expression to:\[4 \cdot 1\]

Step 3: Simplify the Expression

Finally, multiply 4 by 1 to get the simplified result:\[4\]

Key Concepts

Pythagorean Identity

One of the most fundamental relationships in trigonometry is the Pythagorean Identity. At its core, this identity shows the relationship between the square of the cosine and sine functions. It is commonly written as \( an^2 \theta + 1 = \sec^2 \theta\). This identity is very useful for simplifying trigonometric expressions. Let's explain what it means in simpler terms.

Whenever you see \(\sec^2 \theta - \tan^2 \theta\), you should recognize that it equals 1, according to the Pythagorean Identity. This makes it easier to simplify expressions because you can replace this part with 1 in the equation. Utilizing these identities helps in easily managing and manipulating trigonometric expressions without getting tangled up in complex calculations. Always look for opportunities to use the Pythagorean Identity whenever secant and tangent functions are involved in your exercise.

Secant Function

The secant function, often denoted as \(\sec \theta\), is a bit less intuitive than sine or cosine. However, it's equally important in trigonometry. Essentially, the secant function is the reciprocal of the cosine function. Mathematically, this is expressed as \(\sec \theta = \frac{1}{\cos \theta}\). This means that secant is undefined when cosine is zero.

In exercise problems, you often encounter secant squared, \(\sec^2 \theta\). This is simply the square of the secant value. It's frequently used in various trigonometric identities such as the Pythagorean Identity mentioned earlier. Therefore, whenever \(\sec^2 \theta\) appears, consider it as a potential spot where replacing with related identities can simplify the expression.

Understanding secant can feel more challenging at first, but once you link it with cosine, its role in trigonometric identities becomes much more apparent.

Tangent Function

The tangent function, represented as \(\tan \theta\), is another key trigonometric function. It's defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This means it measures the angle of slope between a point on a right triangle and the x-axis. In simpler terms, while sine and cosine are about direct point measurements, tangent is about the angle's inclination.

When you square the tangent function, \(\tan^2 \theta\), its potential for simplifying expressions increases. In the context of our original exercise, notice how \(\tan^2 \theta\) can be replaced using the Pythagorean Identity to make complex simplifications straightforward.

The tangent function is often used alongside secant in problems involving trigonometric identities. Recognizing and utilizing the relationships between these functions is crucial in efficiently solving or simplifying trigonometric expressions.