Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\sec \theta}{\csc \theta}=\tan \theta $$

Step-by-Step Solution

Answer

\(\frac{\sec \theta}{\csc \theta} = \tan \theta\) is an identity because both sides simplify to \(\frac{\sin \theta}{\cos \theta}\).

Step 1: Understand the Identity and Given Equation

We are given the equation \( \frac{\sec \theta}{\csc \theta} = \tan \theta \). The task is to prove that this is an identity, meaning it holds true for all values of \( \theta \) for which functions are defined.

Step 2: Express Trigonometric Functions in Terms of Sine and Cosine

Recall the definitions of the trigonometric functions involved: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Also, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute these into the given equation.

Step 3: Rewrite the Left Side

Substituting the definitions, we get:\[\frac{\sec \theta}{\csc \theta} = \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}}.\]This simplifies to:\[\frac{\sin \theta}{\cos \theta}.\]

Step 4: Compare with Right Side

Notice that the expression \( \frac{\sin \theta}{\cos \theta} \) is identical to \( \tan \theta \). Therefore, the given equation contradicts to:\[\tan \theta = \tan \theta.\]This confirms that the original equation \( \frac{\sec \theta}{\csc \theta} = \tan \theta \) is indeed an identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the core trigonometric identities. It plays an essential role in simplifying and understanding trigonometric expressions. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine function. This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This relationship means that wherever you encounter a secant function in an equation, you can replace it with \( \frac{1}{\cos \theta} \) to help simplify the expression. The secant function can be quite useful in various trigonometric problems as it offers an alternative perspective to view angles and ratios.

Cosecant Function

The cosecant function, represented by \( \csc \theta \), is another important trigonometric identity. Similar to the secant function, the cosecant is defined as the reciprocal of the sine function. In mathematical terms, this is expressed as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Understanding this relationship allows you to transform trigonometric expressions, especially when combining multiple identities. Replacing the cosecant with \( \frac{1}{\sin \theta} \) in an expression can make it easier to match with other parts of the equation or verify identities.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function that can be expressed through the ratios of sine and cosine. The tangent of an angle \( \theta \) can be described as:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This definition forms the basis for many trigonometric proofs and simplifications. By expressing \( \tan \theta \) as a ratio of sine to cosine, you can manipulate it easily within trigonometric equations. In proving identities, recognizing this relationship is crucial, as it offers a straightforward path to simplify and verify expressions.

Sine and Cosine Expressions

Sine and cosine functions are the most basic and crucial functions in trigonometry. They form the building blocks for other trigonometric identities and expressions. Understanding how they interact is essential for manipulating and simplifying complex trigonometric expressions.The sine and cosine functions are defined as follows:

• \( \sin \theta \) - the y-coordinate of the unit circle at angle \( \theta \)
• \( \cos \theta \) - the x-coordinate of the unit circle at angle \( \theta \)

By writing other trigonometric functions in terms of sine and cosine, many complex identities become easier to handle. Furthermore, by transforming trigonometric expressions into their sine and cosine components, it is often easier to verify and prove equations like the identity \( \frac{\sec \theta}{\csc \theta} = \tan \theta \).