Problem 111 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\sec x-\cos x+\csc x-\sin x-\sin x \tan x=\cos x \cot x\)

Step-by-Step Solution

Answer

The given equation is an identity because both sides simplify to \( \cos x \cot x \).

Step 1: Simplify the Left Hand Side

The left hand side (LHS) of the equation is \( \sec x - \cos x + \csc x - \sin x - \sin x \tan x \). Begin by rewriting all trigonometric identities in terms of sine and cosine:- \( \sec x = \frac{1}{\cos x} \)- \( \csc x = \frac{1}{\sin x} \)- \( \tan x = \frac{\sin x}{\cos x} \)So the LHS becomes:\[ \frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \sin x \cdot \frac{\sin x}{\cos x} \]

Step 2: Combine and Simplify Terms

Now, simplify the LHS by combining like terms:\[ \frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \frac{\sin^2 x}{\cos x} \]Combine terms with the same denominator:- Combine \( \frac{1}{\cos x} \) and \( - \frac{\sin^2 x}{\cos x} \):\[ \frac{1 - \sin^2 x}{\cos x} \]Remember, \( 1 - \sin^2 x = \cos^2 x \), so the expression becomes:\[ \frac{\cos^2 x}{\cos x} = \cos x \]

Step 3: Further Simplify the Left Hand Side

Now, add \( \cos x \) to the remaining terms:\[ \cos x - \cos x + \frac{1}{\sin x} - \sin x \] simplifies to:\[ \frac{1}{\sin x} - \sin x \]Now simplify \( \frac{1}{\sin x} - \sin x \) using the common denominator which is \( \sin x \):\[ \frac{1 - \sin^2 x}{\sin x} \]Since \( 1 - \sin^2 x = \cos^2 x \), it reduces to:\[ \frac{\cos^2 x}{\sin x} = \cos x \cot x \] which is identical to the right hand side (RHS) of the original equation.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex equations into simpler forms to make them easier to work with or solve. This often requires you to express trigonometric functions in terms of their basic identities.
A good starting point is substituting trigonometric functions with their equivalent identities, such as replacing \(\sec x\) with \(\frac{1}{\cos x}\), \(\csc x\) with \(\frac{1}{\sin x}\), and \(\tan x\) with \(\frac{\sin x}{\cos x}\).

• These substitutions can often simplify addition and subtraction of terms.
• Combining fractions by finding a common denominator and merging like terms is crucial.
• Utilize Pythagorean identities like \(1 - \sin^2 x = \cos^2 x\) to further simplify terms.

After substitution, combining like terms often becomes more straightforward. In our original exercise, expressions with the same denominator were combined, which led to simpler forms such as replacing \(\frac{1-\sin^2 x}{\cos x}\) with \(\cos x\). Simplification doesn't just make tasks easier—it's also essential for verifying identities.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two different-looking expressions are, in fact, equal to each other. In such exercises, simplifying one or both sides of the equation until they match is the key.
Start with one side of the identity and simplify or manipulate it step by step. You can:

• Apply trigonometric identities to rewrite expressions.
• Factor common terms if possible.
• Simplify fractions and radicals as needed.

The goal is to transform one side to look exactly like the other. In the provided solution, the left-hand side (LHS) was simplified comprehensively until it matched the right-hand side (RHS) of \(\cos x \cot x\). Each transformation or application of an identity brought us closer to expressing the LHS in its desired form. Being systematic and meticulous in rewriting and simplifying is crucial in verifying any identity.

Sine and Cosine Relationship

A deep understanding of the sine and cosine relationship is essential in trigonometry as they form the basis of many identities and expressions. The sine and cosine functions relate through the Pythagorean identity:
\[1 - \sin^2 x = \cos^2 x\]
This identity is transformative in simplifying expressions. It allows you to switch between terms involving sine and cosine easily.

• This relationship assists in simplifying complex trigonometric expressions and verifying identities.
• It is particularly handy when dealing with squared terms and differences such as \(1 - \sin^2 x\).

Recognizing and utilizing these primary trigonometric identities enables more straightforward manipulation of expressions. It can significantly improve the accuracy and efficiency of solving trigonometric problems, as seen in breaking down the exercise where the Pythagorean identity was pivotal for simplification.