Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\cot \theta}{\csc \theta}=\cos \theta $$

Step-by-Step Solution

Answer

The equation \( \frac{\cot \theta}{\csc \theta} = \cos \theta \) is an identity.

Step 1: Recall Trigonometric Identities

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). These will help us simplify the given equation: \( \frac{\cot \theta}{\csc \theta} = \cos \theta \).

Step 2: Substitute Trigonometric Identities

Substitute the identities into the left-hand side of the equation: \( \frac{\cos \theta / \sin \theta}{1/ \sin \theta} \).

Step 3: Simplify the Fraction

Simplify the fraction on the left-hand side by multiplying \( \frac{\cos \theta}{\sin \theta} \) by the reciprocal of \( \frac{1}{\sin \theta} \), which gives \( \cos \theta \times \sin \theta = \cos \theta \).

Step 4: Compare Both Sides

Both sides of the equation simplify to \( \cos \theta \), which shows that \( \frac{\cot \theta}{\csc \theta} = \cos \theta \) is true for all \( \theta \) in the domain.

Key Concepts

Understanding Cotangent

The cotangent, often abbreviated as \( \cot \theta \), is one of the lesser-known trigonometric functions, but it plays a crucial role in many mathematical proofs and expressions. It is defined as the reciprocal of the tangent function. The tangent of an angle \( \theta \) in a right triangle is the ratio of the opposite side over the adjacent side, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Thus, the cotangent is:

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

This means that \( \cot \theta \) is the ratio of the adjacent side to the opposite side in a right triangle. Cotangent is particularly useful in simplifying trigonometric expressions, especially when expressions involve ratios of sine and cosine. Understanding this function helps us manipulate equations to achieve simpler forms.

Exploring Cosecant

Cosecant, represented as \( \csc \theta \), is another reciprocal trigonometric function. It is the reciprocal of the sine function, meaning it is the ratio of the hypotenuse to the opposite side in the context of a right triangle. This can be mathematically expressed as:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} \)

Cosecant is less commonly used than sine, cosine, and tangent, but it is very effective in simplifying complex trigonometric identities. Knowing \( \csc \theta \) allows us to adjust trig expressions by converting sine terms into a form that can be easily managed, just like in our given exercise where it helped simplify the fraction.

Mastering Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a more manageable form, often by using basic trigonometric identities and reciprocal relationships. This process requires practice and a good understanding of the functions and their interactions. Here are a few tips:

• Use known identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) to transform expressions.
• Apply reciprocal identities, such as turning \( \frac{1}{\sin \theta} \) into \( \csc \theta \), to simplify complex fractions.
• Consider multiplying by the conjugate if you need to get rid of denominators in complex formulas.

In the exercise, by applying the reciprocal identities \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \), we can transform and simplify the expression to prove the identity. This simplification makes equations easier to handle and solves complex problems by breaking them into more straightforward parts.