Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$(\sec \theta-\tan \theta)(\csc \theta+1)=\cot \theta$$

Step-by-Step Solution

Answer

The identity is verified as both sides of the expression are equivalent.

Step 1: Rewrite in terms of sine and cosine

We will start by rewriting the identity in terms of sine and cosine. This means that \( \sec \theta \) becomes \( \frac{1}{\cos \theta}\), \( \tan \theta \) becomes \( \frac{\sin \theta}{\cos \theta}\), \( \csc \theta \) becomes \( \frac{1}{\sin \theta}\), and \( \cot \theta \) becomes \( \frac{\cos \theta}{\sin \theta}\). So, the expression becomes \( \left(\frac{1}{\cos \theta}-\frac{\sin \theta}{\cos \theta}\right)\left(\frac{1}{\sin \theta}+1\right) = \frac{\cos \theta}{\sin \theta}\).

Step 2: Simplify the expression

Next, multiply both terms on the left-hand side. This will give four terms and the equation now is: \( \frac{1}{\cos \theta \sin \theta} + \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta \sin \theta} - \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta}{\sin \theta}\). This equation then simplifies to \( \frac{1 - \sin^2 \theta}{ \cos \theta \sin \theta} + \frac{1-\sin \theta}{\cos \theta} = \frac{\cos \theta}{\sin \theta}\). The numerator in the first fraction on the left-hand side is equal to \(\cos^2 \theta\) by Pythagorean identity. Substitute \(1 - \sin^2 \theta\) with \(\cos^2 \theta\) and simplify.

Step 3: Continue simplifying and verify the identity

After replacing \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), we obtain \( \frac{\cos \theta + 1 - \sin \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta}\). As both the left-hand side and the right-hand side are equal, this verifies the identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a fundamental trigonometric function that can be derived from the cosine function. It is defined as the reciprocal of the cosine function, which means:

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

Understanding the secant function is important because it appears in various trigonometric identities and equations. In the identity \((\sec \theta - \tan \theta)(\csc \theta + 1) = \cot \theta\), the secant function translates to \( \frac{1}{\cos \theta} \).
This is used to manipulate expressions algebraically by converting complex trigonometric identities into simpler forms involving sine and cosine. By expressing \( \sec \theta \) in its reciprocal form, it becomes easier to simplify and verify identities such as the one given in the exercise.

Tangent Function

The tangent function, \( \tan \theta \), is another key trigonometric function. It expresses the ratio of the sine function to the cosine function:

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

This function is crucial because it describes the slope of an angle in the unit circle. It also plays a significant role in various trigonometric identities and equations. In the identity provided, \( \tan \theta \) is expressed as \( \frac{\sin \theta}{\cos \theta} \), making it easier to handle within algebraic manipulations.
Transforming \( \tan \theta \) in terms of sine and cosine allows us to perform operations such as adding, subtracting, or multiplying to verify given trigonometric identities. This technique is commonly used in trigonometry problems to simplify expressions.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is the reciprocal of the sine function, which can be expressed as:

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

As with the secant and tangent functions, expressing the cosecant function in terms of its basic components helps simplify complex trigonometric expressions. In the identity \((\sec \theta - \tan \theta)(\csc \theta + 1) = \cot \theta\), \( \csc \theta \) is rewritten as \( \frac{1}{\sin \theta} + 1 \).
This expression allows us to break down and simplify the identity step-by-step. By representing the cosecant function in terms of sine, we can perform necessary mathematical operations to verify and prove trigonometric identities. Thus, understanding the cosecant function helps in tackling problems with similar mathematical structures.

Cotangent Function

The cotangent function, \( \cot \theta \), is the reciprocal of the tangent function pronounced through its relation:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

In the given identity problem, \( \cot \theta \) is provided as a simplified target form to prove the correctness of the equation. The cotangent function is commonly used in verifying identities by rephrasing complex expressions into straightforward terms.
Converting \( \cot \theta \) in terms of sine and cosine makes it more manageable to compare both sides of a trigonometric equation and confirm the equality. The cotangent, with its expression in terms of basic trigonometric ratios, plays a crucial part in evaluating and simplifying mathematical expressions in trigonometry.