Problem 6 - Algebra 2

Question

Simplify each expression. $$ \sec ^{2} \theta-1 $$

Step-by-Step Solution

Answer

\( \tan^2 \theta \)

Step 1: Identify a Trigonometric Identity

Recall the Pythagorean identity related to the secant function. The identity states that \( \sec^2 \theta - 1 = \tan^2 \theta \).

Step 2: Apply the Identity

Substitute \( \sec^2 \theta - 1 \) using the Pythagorean identity from Step 1, simplifying it to \( \tan^2 \theta \). Thus, the expression \( \sec^2 \theta - 1 \) simplifies directly to \( \tan^2 \theta \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, and it's essential for simplifying many trigonometric expressions. It relates three of the primary trigonometric functions, sine, cosine, and tangent. The basic form of the Pythagorean Identity is given by:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity can be manipulated to derive other identities. For functions involving the secant and tangent, we utilize the identity:\[\sec^2 \theta - 1 = \tan^2 \theta\]This version of the identity is particularly helpful when working with problems involving these two functions. Remember, the secant is the reciprocal of cosine, and the tangent is the ratio of sine to cosine. Whenever you see an expression with \(\sec^2 \theta\), this identity can simplify your work considerably.

Secant Function

The secant function is one of the six fundamental trigonometric functions, denoted as \(\sec \theta\). It is defined as the reciprocal of the cosine function. This means:\[\sec \theta = \frac{1}{\cos \theta}\]Understanding the secant function is important because it appears frequently in trigonometric identities and problems, like in the Pythagorean identity we discussed earlier.

• The secant function has a range of \((-\infty, -1] \cup [1, \infty)\).
• It is undefined where cosine is zero, which occurs at odd multiples of \(\frac{\pi}{2}\).

The secant function also helps determine relationships between other trigonometric functions like tangent, as seen in the identity \(\sec^2 \theta - 1 = \tan^2 \theta\). Knowing these relationships makes solving trigonometric equations much more manageable.

Tangent Function

The tangent function, denoted as \(\tan \theta\), is another essential trigonometric function and is defined as the ratio of the sine function to the cosine function:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]The tangent function is useful in a wide variety of applications, from solving right triangles to modeling periodic phenomena. Understanding the tangent function's behavior and relationships with other functions is crucial for simplifying expressions and solving trigonometric problems.

• The tangent function has a range of all real numbers \((-\infty, \infty)\).
• It is undefined wherever the cosine function is zero, as division by zero is undefined.
• Its periodicity is \(\pi\), meaning \(\tan(\theta + \pi) = \tan \theta\).

The tangent function is closely linked with the secant function through the Pythagorean Identity \(\sec^2 \theta - 1 = \tan^2 \theta\). This relationship is particularly useful when you need to rewrite or simplify trigonometric expressions.