Problem 24 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{1-\sin \theta}{\cos \theta}=\sec \theta-\tan \theta\)

Step-by-Step Solution

Answer

The original equation has been verified. By replacing sec and tan with their equivalent terms and combining the terms on the right side of the equation, we have shown that the left side is equivalent to the right side.

Step 1: Rewrite sec and tan in terms of sin and cos

Rewrite the right side of the equation using the identities \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This gives us: \(\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}\)

Step 2: Make the denominators the same

Combine the two fractions on the right side over a common denominator \( \cos \theta \). This will give us: \(\frac{1 - \sin \theta}{\cos \theta}\)

Step 3: Compare the two sides

Compare the transformed right side of the equation to the left side. We see that they are indeed the same: \(\frac{1-\sin \theta}{\cos \theta}=\frac{1-\sin \theta}{\cos \theta}\)

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is commonly related to the cosine function. Its definition is quite simple:

• Secant is the reciprocal of the cosine function.

To express it mathematically, you can write:\[ \sec \theta = \frac{1}{\cos \theta} \]Understanding the secant function involves knowing that it can never be zero. This is because it is defined as the reciprocal of cosine, and for the secant to be zero, the cosine would need to be undefined, posing a problem for the trigonometric identity itself.
The secant function gives insight into the graph's behavior of cosine, particularly where \( \cos \theta \) tends to small values near their minimum or maximum ranges.

Tangent Function

The tangent function, represented as \( \tan \theta \), is another key player in trigonometry. It is often expressed in terms of sine and cosine functions:

• Tangent is the ratio of sine to cosine.

Mathematically, this is expressed as:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]An interesting aspect of the tangent function is that it becomes undefined whenever \( \cos \theta = 0 \), because division by zero is not possible, leading to vertical asymptotes in its graph.
The tangent function is periodic with a period of \( \pi \), and it plays a significant role in defining angles' slopes and relationships in trigonometry. Understanding these relationships helps when working with identities and transformations.

Verifying Identities

Trigonometric identities are equations involving trig functions that hold true for all values of the involved variables for which both sides of the equation are defined. Verifying identities is a process where you prove that two expressions are equivalent.

• Work on one side of the equation at a time.
• Simplify or transform using known identities.

In the given exercise, you need to verify that:\[ \frac{1-\sin \theta}{\cos \theta} = \sec \theta - \tan \theta \]Start by transforming the right side using known expressions for secant and tangent in terms of sine and cosine. By making common denominators, it's easier to compare both sides of the equation and prove equality. Verifying identities develops problem-solving skills and deepens understanding of trigonometric functions.