Problem 8 - Algebra 2

Question

Verify each identity. $$ \cot \theta=\frac{\csc \theta}{\sec \theta} $$

Step-by-Step Solution

Answer

The given trigonometric identity is indeed true. Both sides simplify to the same value, thus verifying the identity.

Step 1: Break Down the Left Side

To make the verification process simpler, we first express the trigonometric function \(\cot \theta\) in terms of sine and cosine, as they are the basic trigonometric functions. Hence, \( \cot \theta=\frac{\cos \theta}{\sin \theta}\).

Step 2: Break Down the Right Side

Next, we express the functions \(\csc \theta\) and \(\sec \theta\) on the right side of the identity in terms of sine and cosine. Hence, \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta= \frac{1}{\cos \theta}\). Substituting these values, we get the right side to be \(\frac{\csc \theta}{\sec \theta} = \frac{1}{\sin \theta} \times \cos \theta\).

Step 3: Comparing both sides

Comparing the left side of the identity, \(\frac{\cos \theta}{\sin \theta}\), with the right side, \(\frac{1}{\sin \theta} \times \cos \theta\), we can see that they are equal.

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot \theta \), is one of the six fundamental trigonometric functions. It is closely related to the tangent function and is defined as the reciprocal of the tangent. In mathematical terms, \( \cot \theta = \frac{1}{\tan \theta} \).
Alternatively, cotangent can be expressed in terms of sine and cosine. This relationship is particularly useful when working to verify trigonometric identities. The expression is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
This makes the cotangent function an essential tool for converting complex trigonometric expressions involving the tangent function into simpler expressions involving sine and cosine.

Cosecant

The cosecant function, represented as \( \csc \theta \), is another core trigonometric function. It is the reciprocal of the sine function. That is, \( \csc \theta = \frac{1}{\sin \theta} \).
The cosecant is especially vital in trigonometric identities and equations when sine is part of the expression, as it allows for simplification and transformation.

• Understanding \( \csc \theta \) helps in solving problems where inverse relationships involving sine functions occur.
• It also appears frequently in calculus and physics, where it helps in deriving integrations and solutions involving sine.

Secant

The secant function is denoted by \( \sec \theta \) and is defined as the reciprocal of the cosine function. Thus, \( \sec \theta = \frac{1}{\cos \theta} \).
Much like the cosecant, the secant function is useful when dealing with complex trigonometric identities. It simplifies expressions that involve cosines and helps in verifying the identities by converting cosine terms into simpler multiplicative inverses.
Secant is pivotal in analytical geometry and calculus, particularly in problems that require division by cosine or transformation using cosine relationships.

Sine and Cosine

Sine and cosine are two of the most fundamental and widely used trigonometric functions, represented by \( \sin \theta \) and \( \cos \theta \) respectively.
These functions are based on the angles of a right triangle or the unit circle, representing the y-coordinate and x-coordinate of a point on a circle of radius one.

• \( \sin \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse.
• \( \cos \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse.

Sine and cosine are used extensively in various fields such as physics, engineering, and computer graphics for modeling periodic phenomena, among many other applications.
Understanding sine and cosine relations helps in converting other trigonometric functions to basic terms, laying the foundation for solving more complex trigonometric problems.