Problem 42 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\tan 2 \theta+\cot 2 \theta}{\sec 2 \theta}=\csc 2 \theta\)

Step-by-Step Solution

Answer

Our simplified left hand side matches the right hand side. Hence, \(\frac{\tan 2 \theta+\cot 2 \theta}{\sec 2 \theta}=\csc 2 \theta\) is indeed true, assuming \(0 < \theta < 90^\circ\).

Step 1: Convert Identities

The first step is to convert all the trigonometric functions on the LHS to their basic sine and cosine equivalents. Therefore, \(tan(2\theta)\) becomes \(\frac{sin(2\theta)}{cos(2\theta)}\) and \(cot(2\theta)\) is \(\frac{cos(2\theta)}{sin(2\theta)}\), and \(sec(2\theta)\) is \(\frac{1}{cos(2\theta)}\). Substituting these in the original equation we get, \(\frac{\frac{sin(2\theta)}{cos(2\theta)}+\frac{cos(2\theta)}{sin(2\theta)}}{\frac{1}{cos(2\theta)}}\)

Step 2: Simplify the Fraction

Multiply the numerator and the denominator by \(cos(2\theta)\) to get rid of the fractions within the fraction. This yields \(\frac{sin(2\theta) + cos^2(2\theta)}{1}\)

Step 3: Use the Pythagorean Identity

We should recognize this expression from the Pythagorean identity \(sin^2(x) + cos^2(x)\) =1. Since we have \(sin^2(2\theta) + cos^2(2\theta)\) in the numerator, it simplifies to 1

Key Concepts

Verifying Trigonometric Identities

Verifying trigonometric identities can be likened to proving that two different expressions are indeed the same from a mathematical standpoint. It involves a systematic approach to manipulate and transform the expressions using trigonometric identities so that both sides of the equation match. In our given exercise, \

\(\frac{\tan 2 \theta+\cot 2 \theta}{\sec 2 \theta}=\csc 2 \theta\)

, verification begins by expressing the functions in terms of sine and cosine which are the basic building blocks for all trigonometric functions.

Converting Trigonometric Functions

In trigonometry, converting functions into basic ones—sine and cosine—is a pivotal strategy for simplification. It's essential for verifying identities as well as solving complex equations.

Take the example from our exercise. The tangent function, represented by \(\tan(2\theta)\), is rewritten as \(\frac{\sin(2\theta)}{\cos(2\theta)}\). Similarly, cotangent \(\cot(2\theta)\) is the reciprocal of tangent and thus converts to \(\frac{\cos(2\theta)}{\sin(2\theta)}\), and secant \(\sec(2\theta)\), being the reciprocal of cosine, changes to \(\frac{1}{\cos(2\theta)}\). This step is imperative for further simplification of trigonometric expressions.

Pythagorean Trigonometric Identity

The Pythagorean trigonometric identity is fundamental to trigonometry, stating that \(\sin^2(x) + \cos^2(x) = 1\) for any angle x. These identities are derived from the Pythagorean Theorem and relate to the unit circle.

In the context of the given problem, after converting and simplifying the given expression, the numerator of our equation resembles the Pythagorean identity, which allows us to replace \(\sin^2(2\theta) + \cos^2(2\theta)\) with 1. This understanding plays a crucial role in verifying complex trigonometric identities.

Simplifying Trigonometric Expressions

The process of simplifying trigonometric expressions involves algebraic manipulation, including factoring, expanding, and canceling, with the primary goal of reducing expressions to their simplest forms. Simplifying these expressions requires knowledge of trigonometric identities and the skill to discern which identity to apply in order to facilitate the simplification process.

In our exercise, once the trigonometric functions are converted to sine and cosine terms, simplification is achieved by multiplying both the numerator and the denominator by \(\cos(2\theta)\) to eliminate the compound fraction. This tactic is often employed to streamline the verification of trigonometric identities.