Problem 20 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{1}{\tan \beta} + \tan \beta = \dfrac{\sec^2 \beta}{\tan \beta} \)

Step-by-Step Solution

Answer

Yes, the identity \(\dfrac{1}{\tan \beta} + \tan \beta = \dfrac{\sec^2 \beta}{\tan \beta}\) is valid.

Step 1: Convert to sine and cosine

First, convert the \(\tan \beta\) and \(\sec \beta\) into \(\sin \beta\) and \(\cos \beta\) terms respectively. Hence, \(\dfrac{1}{\tan \beta} + \tan \beta\) becomes \(\dfrac{\cos \beta}{\sin \beta} + \dfrac{\sin \beta}{\cos \beta}\) and \(\dfrac{\sec^2 \beta}{\tan \beta}\) becomes \(\dfrac{1}{\cos^2 \beta} * \dfrac{\cos \beta}{\sin \beta}\)

Step 2: Simplify the fractions

Next, simplify the fractions. The left hand side becomes \(\dfrac{cos^2 \beta + sin^2 \beta}{\cos \beta \sin \beta}\). For the right hand side, the \(\cos \beta\) in the numerator and denominator will cancel out, leaving \(\dfrac{1}{\cos \beta \sin \beta}\)

Step 3: Apply Trigonometric Identity

On the left hand side, apply the trigonometric identity \(sin^2 \beta + cos^2 \beta = 1\). Now, both sides of the equation are equal, thus proving the identity is valid.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for solving a wide range of mathematical problems, including those related to trigonometric functions.
Trigonometric identities are equalities that involve trigonometric functions and are true for all values of the involved variables where both sides of the equality are defined. These identities are helpful not only in geometric and trigonometric calculations but also in simplifying complex expressions and solving equations.

Why are They Important?

They allow us to transform one trigonometric function into another, revealing a relationship that can make problem-solving more straightforward. For instance, the identity \(sin^2 \theta + cos^2 \theta = 1\) is one of the most fundamental identities, known as the Pythagorean identity. It's essential for changing expressions from one form to another to facilitate easier calculation or comparisons.

• They simplify complicated expressions.
• They are used to prove whether a given equation is an identity.
• They assist in solving trigonometric equations.

In the example provided, we use trigonometric identities to transform and simplify the original equation, eventually verifying the given identity.

Sine and Cosine

The sine and cosine functions are foundational to trigonometry. Their significance comes from their relationship to right-angled triangles, as well as their periodic nature, which is useful in describing oscillatory motions.
The sine of an angle in a right-angled triangle is defined as the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse.

Relation to the Unit Circle

In the unit circle, sine and cosine correspond to the y-coordinate and x-coordinate of a point on the circle, respectively. This definition extends their use beyond just acute angles to all real numbers, facilitating their use in various fields such as physics, engineering, and signal processing.

• Sine and cosine are periodic functions with a period of \(2\pi\).
• They have range values between -1 and 1.

When verifying trigonometric identities like in the provided exercise, converting tan and sec to their sine and cosine components is a common strategy to simplify and solve the problem.

Secant and Tangent Functions

Alongside sine and cosine, the secant and tangent functions play a significant role in trigonometry.

Defining Tangent and Secant

The tangent of an angle is the ratio of the sine to the cosine of that angle: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). The secant is the reciprocal of the cosine: \(\sec \theta = \frac{1}{\cos \theta}\). These functions are important when dealing with angles beyond the first quadrant because they can be used to model phenomena that exhibit certain types of symmetry or periodicity.

• The tangent function has a period of \(\pi\) and is undefined whenever cosine is zero (at odd multiples of \(\pi/2\)).
• The secant function has the same period as cosine, \(2\pi\), and is undefined whenever cosine is zero.

By understanding these relationships, as illustrated in the exercise solution, students can manipulate trigonometric expressions to verify identities, build upon them, and even derive additional identities.