Problem 6 - Algebra 2

Question

Verify each identity. $$ \tan \theta \cot \theta=1 $$

Step-by-Step Solution

Answer

The given trigonometric identity \( \tan \theta \cdot \cot \theta = 1 \) has been verified and is correct.

Step 1: Define the Trigonometric Terms

Start by understanding and defining the given trigonometric terms. Here, the terms are tan(θ) and cot(θ). By definition, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

Step 2: Substitute the Definitions

Next, substitute the definitions of tan(θ) and cot(θ) into the given identity and simplify. Thus, \( \tan \theta \cdot \cot \theta = \frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} \).

Step 3: Simplify

In this step, simplify the result from the previous step. The \( \sin \theta \) in the nominator and denominator will cancel out, and likewise the \( \cos \theta \) values in the nominator and denominator will also cancel out, giving 1, thus proving the identity.

Key Concepts

Tangent and Cotangent

Tangent and cotangent are two important trigonometric functions that often appear together in mathematical problems. Both functions are related to the sine and cosine functions.
The tangent of an angle ( \( \theta \) ) is defined as the ratio of the sine to the cosine of that angle, expressed by the formula:

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

Cotangent, on the other hand, is the reciprocal of the tangent function. You can express cotangent as:

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

These definitions show how cotangent and tangent are closely interrelated and how they involve sine and cosine. Understanding these relationships is key when working with identities like \( \tan \theta \cdot \cot \theta = 1 \). Recognizing that the product of tangent and cotangent simplifies to one helps in verifying such identities.

Simplifying Expressions

Simplifying trigonometric expressions can make solving equations or verifying identities much easier. In our example, we'll work through simplifying the expression \( \tan \theta \cdot \cot \theta \).
First, substitute the tangent and cotangent definitions we discussed earlier:

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

Now, multiply these expressions:

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

The sine and cosine terms both cancel out because they appear in both the numerator and the denominator:

• This results in \( 1 \)

This simplification demonstrates why \( \tan \theta \cdot \cot \theta = 1 \) is a valid trigonometric identity. Simplifying expressions by canceling common terms is a powerful tool in mathematics.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in geometry and calculus. They describe relationships between angles and sides of triangles. The basics include three primary functions: sine, cosine, and tangent. These are accompanied by their reciprocal functions: cosecant, secant, and cotangent.
The main trigonometric functions are defined as follows:

• Sine \( (\sin \theta) \) - the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine \( (\cos \theta) \) - the ratio of the adjacent side to the hypotenuse.
• Tangent \( (\tan \theta) \) - the ratio of the opposite side to the adjacent side.

The reciprocal functions include:

• Cosecant \( (\csc \theta) = \frac{1}{\sin \theta} \)
• Secant \( (\sec \theta) = \frac{1}{\cos \theta} \)
• Cotangent \( (\cot \theta) = \frac{1}{\tan \theta} \)

These functions are used to model periodic phenomena like waves. Understanding these fundamental definitions helps when working with more complex trigonometric identities and functions.