Problem 26 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\tan \theta+\cot \theta=2 \csc 2 \theta\)

Step-by-Step Solution

Answer

The identity is verified using trigonometric identities and simplifications.

Step 1: Express Tangent and Cotangent

Start by expressing \( \tan \theta \) and \( \cot \theta \) in terms of sine and cosine.\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 2: Combine the Expressions

Combine the expressions from Step 1 over a common denominator:\[ \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

Step 3: Apply Pythagorean Identity

Apply the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):\[ \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \]

Step 4: Express as Cosecant and Double Angle

Recognize that \( \frac{1}{\sin \theta \cos \theta} \) can be rewritten using the double angle identity:\[ \sin 2\theta = 2 \sin \theta \cos \theta \]Thus,\[ \frac{1}{\sin \theta \cos \theta} = \frac{2}{2 \sin \theta \cos \theta} = \frac{2}{\sin 2\theta} = 2 \csc 2 \theta \]

Step 5: Verify the Identity

Now, observe that both sides of the original equation:\[ \tan \theta + \cot \theta = 2 \csc 2\theta \]are indeed equal, thus confirming the identity.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine of the angle \( \theta \). Formally, this relationship is expressed as:

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

This function is significant in trigonometry as it relates to the slope of an angle when seen on a unit circle. In broader terms, it is used to determine angles and distances in various branches of science and engineering.
Understanding the tangent function's computation in terms of sine and cosine is crucial when simplifying trigonometric expressions. This conceptualization can aid in verifying complex identities such as \( \tan \theta + \cot \theta = 2 \csc 2 \theta \), as it forms the first step in expressing and combining trigonometric functions.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is another vital trigonometric function that acts as the reciprocal of the tangent function. It is mathematically described by:

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

This function is particularly essential when it comes to division-type problems in trigonometry, offering an alternative perspective by flipping the ratio between sine and cosine.
In the context of verifying trigonometric identities, understanding the cotangent function's expression as a reciprocal makes it easier to unite it with the tangent function, as undertaken in the equation \( \tan \theta + \cot \theta = 2 \csc 2 \theta \). This reciprocal relationship provides an avenue to solve complex algebraic trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a cornerstone relation in trigonometry that connects the sine and cosine functions: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity reflects the fundamental geometric principles behind a right triangle, where the sum of the squares of a right triangle's legs (sine and cosine) equals the square of the hypotenuse (1).
Applying this identity is useful in simplifying expressions involving sine and cosine. For example, in the step \( \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \), employing the Pythagorean identity replaces \( \sin^2 \theta + \cos^2 \theta \) with 1. This simplification is crucial for reducing complex expressions to more manageable forms, aiding the overall process of verifying trigonometric identities.

Double Angle Identity

The double angle identity for sine is expressed as: \( \sin 2\theta = 2 \sin \theta \cos \theta \). This identity is pivotal in transforming products of sine and cosine into usable trigonometric functions of a single angle.
Applying the double angle identity, such as in \( \frac{1}{\sin \theta \cos \theta} = \frac{2}{\sin 2\theta} \), allows the reformulation of complex expressions into a simpler sine function with a doubled angle. In the given problem, recognizing and employing the double angle identity helps verify that \( \tan \theta + \cot \theta \) equals \( 2 \csc 2 \theta \).

• The double angle identity simplifies analysis of trigonometric expressions.
• It creates pathways to prove and verify complex equations by reducing them to known identities.

Understanding this identity not only simplifies calculations but also underscores the interconnectivity among various trigonometric functions.