Problem 33 - Algebra 2

Question

Simplify each trigonometric expression. $$ \csc \theta-\cos \theta \cot \theta $$

Step-by-Step Solution

Answer

The simplified expression of \(\csc \theta - \cos \theta \cot \theta\) is \(0\)

Step 1: Substitute the Trigonometric Identities

Replace \(\csc \theta\) with \(1/\sin \theta\) and \(\cot \theta\) with \(\cos \theta / \sin \theta\), as these are their respective reciprocal identities. Thus, the given expression becomes: \(1/\sin \theta - \cos \theta \times (\cos \theta / \sin \theta)\)

Step 2: Simplify the Expression

The \(\cos \theta\) in the second term of the formula will cancel out when multiplied by another \(\cos \theta\), leaving \(-1/ \sin \theta\). The expression can now be simplified to: \(1/\sin \theta - 1/\sin \theta\)

Step 3: Combine the Like Terms

The \(1/\sin\theta\) in both terms cancels each other out, hence the final simplified expression is: \(0\)

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex trigonometric terms into simpler forms. This helps in solving equations or identifying patterns. The goal is to rewrite expressions using basic trigonometric identities to make them easier to work with.
To simplify an expression:

• Identify known trigonometric identities that fit the expression.
• Substitute these identities into the expression.
• Perform algebraic simplifications like combining like terms or factoring.

In our exercise, the expression \( \csc \theta - \cos \theta \cot \theta \) is simplified by using trigonometric identities, ultimately breaking it down to 0. This shows the power of recognizing and applying identities in simplifying tasks.

Reciprocal Identities

Reciprocal identities are essential to understanding and working with trigonometric expressions. They express trigonometric functions as reciprocals of one another, allowing flexibility in calculations and simplifications.
Some basic reciprocal identities include:

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

In the given exercise, we replace \( \csc \theta \) with \( \frac{1}{\sin \theta} \) and \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \).
This makes it possible to unify terms under a common denominator, paving the way for further simplification.

Cotangent and Cosecant Functions

The cotangent and cosecant functions are closely linked through reciprocal identities and are often used in various trigonometric transformations.

• **Cosecant (\( \csc \theta \)):** The cosecant function is the reciprocal of the sine function, represented as \( \csc \theta = \frac{1}{\sin \theta} \).
• **Cotangent (\( \cot \theta \)):** The cotangent function is the reciprocal of the tangent function or equivalently \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

These functions help in expressing complex trigonometric constructs in simpler terms.
In our expression, by recognizing these relationships, we replaced trigonometric functions with their reciprocals. This enabled us to easily manipulate and reduce the expression to zero.