Problem 19 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \frac{1+\csc x}{\cos x+\cot x} $$

Step-by-Step Solution

Answer

\(\sec x\)

Step 1: Rewrite in terms of sin and cos

\(\frac{1+\csc x}{\cos x + \cot x} = \frac{1 + \frac{1}{\sin x}}{\cos x + \frac{\cos x}{\sin x}} = \frac{\frac{\sin x + 1}{\sin x}}{\frac{\cos x \sin x + \cos x}{\sin x}}\)

Step 2: Simplify

\(= \frac{\sin x + 1}{\cos x(\sin x + 1)} = \frac{1}{\cos x} = \sec x\)

Step 3: State the result

\(\frac{1+\csc x}{\cos x + \cot x} = \sec x\)

Key Concepts

Cosecant Function

The cosecant function, denoted as \(\csc x\), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. In other words, \(\csc x = \frac{1}{\sin x}\). This means that the cosecant function is undefined when the sine of \(x\) is zero, as division by zero is not possible.
\(\csc x\) is particularly useful in cases where you need to express angles in terms of their sine but involve division. It often appears in trigonometric identities and helps to simplify expressions and solve equations where sine might lead to complexities.
By using \(\csc x\), we can transform parts of a trigonometric expression to make further simplifications possible. It is important to have a good understanding of how it interacts with other trigonometric functions, especially in more involved mathematical expressions.

Cotangent Function

The cotangent function, represented by \(\cot x\), is another key trigonometric function. It is defined as the reciprocal of the tangent function. Specifically, \(\cot x = \frac{1}{\tan x}\). In terms of sine and cosine, \(\cot x\) can also be expressed as \(\frac{\cos x}{\sin x}\).
This dual way of expressing \(\cot x\) makes it a versatile function when simplifying trigonometric expressions. Like \(\csc x\), \(\cot x\) is undefined when its denominator is zero; in this case, when \(\sin x\) is zero.
The cotangent function often emerges in trigonometric identities, where it can assist in reducing equations to more manageable forms. Its relationship with tangent also links it to the other circular functions, giving it a pivotal role in solving trigonometric problems and identities.

Simplifying Expressions

Simplifying trigonometric expressions involves rewriting them in a simpler or more readable form. The goal is often to reduce complexity and make further mathematical operations easier.
To simplify expressions like \(\frac{1+\csc x}{\cos x + \cot x}\), understanding of the reciprocal relationships—like those of \(\csc x\) and \(\cot x\)—is essential. By substituting the reciprocals \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\), the expression can be rewritten in terms of sine and cosine, which are typically easier to manipulate.
Using trigonometric identities and reciprocal relationships can lead to further simplifications, such as combining terms or canceling out parts of the expression. Practicing these techniques helps develop a skill set that makes solving trigonometric problems and performing algebraic operations more straightforward.