Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity \( \tan \theta + \cot \theta = 2 \csc 2\theta \) is correct.

Step 1: Rewrite in terms of Sine and Cosine

First, express \( \tan \theta \) and \( \cot \theta \) in terms of sine and cosine:\[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]Therefore, the left-hand side becomes:\[ \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \]

Step 2: Find a Common Denominator

To combine the fractions, find a common denominator:\[ \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

Step 3: Simplify Using Pythagorean Identity

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

Step 4: Rewrite Using Double Angle Identity

Use the identity for the double angle for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \).Thus, \( \frac{1}{\sin \theta \cos \theta} = \frac{2}{2 \sin \theta \cos \theta} = \frac{2}{\sin 2\theta} \).This is equivalent to \( 2 \csc 2 \theta \), since \( \csc 2\theta = \frac{1}{\sin 2\theta} \).

Step 5: Confirm Identity

We have shown that:\[ \tan \theta + \cot \theta = 2 \csc 2\theta \]Thus, the identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental identities in trigonometry, providing a relationship between sine and cosine. This identity states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity arises from the Pythagorean theorem applied to a unit circle. On a unit circle, any coordinate makes a right triangle with the radius, yielding this identity from the theorem that for a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
In practical terms, the Pythagorean identity is useful in simplifying trigonometric expressions. For example, in the exercise solution, the expression \( \sin^2 \theta + \cos^2 \theta \) simplifies to 1. This significantly facilitates the solution, converting complex expressions into simpler forms, thereby making calculations more manageable.

sine and cosine

Sine and cosine are foundational to trigonometry, comprising the primary functions that describe periodic phenomena. The sine function, \( \sin \theta \), measures the vertical component of a point on the unit circle, while cosine, \( \cos \theta \), gives the horizontal component. Together, they span values from -1 to 1 as the angle \( \theta \) travels around the circle.
Their reciprocal and quotient functions are crucial for solving trigonometric identities and equations. In the given exercise:

• Tangent, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), arises from the ratio of sine to cosine.
• Cotangent, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), is the reciprocal of tangent.
• Cosecant, \( \csc \theta = \frac{1}{\sin \theta} \), and secant, \( \sec \theta = \frac{1}{\cos \theta} \), are reciprocals of sine and cosine respectively.

This detailed understanding of how these functions interact allows one to manipulate and verify trigonometric identities.

double angle identity

The double angle identity provides a technique to express trigonometric functions of double angles in terms of single angles. One of the most useful versions of this identity is for sine, expressed as \( \sin 2\theta = 2 \sin \theta \cos \theta \). This takes advantage of both sine and cosine functions to represent twice the angle in a simpler manner.
In this exercise, the double angle identity helps transform the expression \( \frac{1}{\sin \theta \cos \theta} \) into \( \frac{2}{\sin 2\theta} \). This transformation is crucial to match the right-hand side of the identity \( 2 \csc 2\theta \), where \( \csc 2\theta = \frac{1}{\sin 2\theta} \). Understanding this identity allows you to manipulate and simplify expressions, ensuring trigonometric problems become easier to approach and proving identities becomes more straightforward.