Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\sec x\).

Step 1: Rewrite Cosecant and Cotangent

The first step is to rewrite the terms \(\csc x\) and \(\cot x\) in terms of sine and cosine. Recall that \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substitute these into the expression:\[\frac{1 + \frac{1}{\sin x}}{\cos x + \frac{\cos x}{\sin x}}\]

Step 2: Simplify the Denominator

Let's simplify the denominator \(\cos x + \frac{\cos x}{\sin x}\). Find a common denominator for the terms in the denominator:\[\cos x + \frac{\cos x}{\sin x} = \frac{\cos x \sin x}{\sin x} + \frac{\cos x}{\sin x} = \frac{\cos x \sin x + \cos x}{\sin x}\]

Step 3: Simplify the Numerator

Now simplify the numerator \(1 + \frac{1}{\sin x}\). Similarly, find a common denominator:\[1 + \frac{1}{\sin x} = \frac{\sin x}{\sin x} + \frac{1}{\sin x} = \frac{\sin x + 1}{\sin x}\]

Step 4: Simplify the Fraction

Substitute the simplified numerator and denominator back into the expression:\[\frac{\frac{\sin x + 1}{\sin x}}{\frac{\cos x(\sin x + 1)}{\sin x}}\]When simplifying a fraction of fractions, multiply by the reciprocal of the denominator:\[\frac{\sin x + 1}{\sin x} \times \frac{\sin x}{\cos x(\sin x + 1)}\]Cancel out the common terms \(\sin x\) and \(\sin x + 1\):\[\frac{1}{\cos x}\]This is equivalent to \(\sec x\).

Step 5: Final Answer

Now, we can state the final simplified expression. The original expression simplifies to:\[\sec x\]

Key Concepts

Cosecant

Cosecant is a trigonometric function, and it is the reciprocal of the sine function. It is commonly represented as \(\csc x\). In mathematical terms, this can be written as:

• \(\csc x = \frac{1}{\sin x}\)

Understanding the reciprocal nature of cosecant helps simplify expressions that involve \(\sin x\). In our exercise, we took \(1 + \csc x\) and rewrote \(\csc x\) as \(\frac{1}{\sin x}\) to help simplify the expression.
This substitution turns complex trigonometric terms into forms that are easier to work with, especially when dealing with algebraic operations like finding a common denominator.

Cotangent

Cotangent, like cosecant, is another fundamental trigonometric function. It is the reciprocal of the tangent function and can be expressed using sine and cosine. Cotangent is usually denoted as \(\cot x\), and mathematically, it is represented as:

• \(\cot x = \frac{\cos x}{\sin x}\)

In the simplification process of trigonometric expressions, translating \(\cot x\) into its sine and cosine components aids in aligning terms under a common denominator.
In this exercise, rewriting \(\cot x\) in terms of sine and cosine was a crucial step. This conversion allowed us to simplify the denominator of the given complex fraction. Recognizing connections among trigonometric functions often streamlines solving these expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved angles. They play a substantial role in simplifying complex trigonometric expressions. Some key identities are:

• Reciprocal identities, like \(\csc x = \frac{1}{\sin x}\)
• Symmetrical identities, such as \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\)
• Pythagorean identities, for instance, \(\sin^2 x + \cos^2 x = 1\)

These identities allow us to rewrite and reduce expressions to their simplest forms by unveiling hidden relationships between the functions.
In our problem, applying trigonometric identities helped reframe the initial expression using fundamental trigonometric relationships. This transformation facilitated the cancellation of terms, leading to a straightforward expression, \(\sec x\), as the final simplified result.