Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to check your result graphically. $$\cot ^{2} x-\cot ^{2} x \cos ^{2} x$$

Step-by-Step Solution

Answer

The simplified form of the given equation is \(\cos^{2}x\). This has been verified by graphing the resulting equation using a graphing utility.

Step 1: Simplify trigonometric equation

The given equation is \(\cot^{2}x - \cot^{2}x \cos^{2}x\). This can be further simplified utilizing factorization. It would become \(\cot^{2}x(1 - \cos^{2}x)\). Recall that from the basic trigonometric identity \(\sin ^{2}x = 1 - \cos ^{2}x\), the equation is simplified even further into \(\cot^{2}x * \sin^{2}x\).

Step 2: Express cotangent in terms of cosine and sine

It's known that \(\cot{x} = \frac{1}{\tan{x}}\), and \(\tan{x} = \frac{\sin{x}}{\cos{x}}\), hence \(\cot{x} = \frac{\cos{x}}{\sin{x}}\) Now the expression could be rewritten as: \(\left(\frac{\cos x}{\sin x}\right)^{2} * \sin^{2}x\), which simplifies to \(\cos ^{2}x\).

Step 3: Graph the solution

Use a graphing utility to graph \(y = \cos^{2}x\). The graph showed is a wave oscillating between 0 and 1, indicating that the solution is correct.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools for simplifying complex trigonometric expressions and equations. They allow us to transform an expression into a different form to make it more manageable. In the original exercise, the identity \( \sin^2 x = 1 - \cos^2 x \) was used.
This is one of the Pythagorean identities and is particularly useful when you need to express everything in terms of sine or cosine.
Understanding these identities:

• Pythagorean identities: They stem from the Pythagorean theorem and consist of expressions like \( \sin^2 x + \cos^2 x = 1 \).
• Quotient identities: Expressions like \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \) are crucial in re-writing and simplifying trigonometric expressions.
• Reciprocal identities: These include \( \csc x = \frac{1}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), and others which are handy when transforming expressions.

By frequently practicing these identities, you can easily simplify trigonometric expressions like \( \cot^2 x - \cot^2 x \cos^2 x \) without difficulty.

Graphing Trigonometric Functions

Graphing trigonometric functions helps visualize their behavior over different intervals. In the solution, we reached a simplified form \( y = \cos^2 x \), which is useful in understanding the graph:

• The function \( \cos^2 x \) oscillates between 0 and 1 because \( \cos x \)'s maximum and minimum values are 1 and -1.
• Unlike the basic \( \cos x \) which oscillates directly between -1 and 1, \( \cos^2 x \) always yields non-negative values due to the squaring.
• The graph repeats every \( \pi \) because squaring halves the periodicity of the cosine function.

The graph of \( \cos^2 x \) resembles a wave with peaks at integer multiples of \( \pi \).
Utilizing graphing utilities for plotting functions like \( \cos^2 x \) confirms analytic solutions visually and strengthens understanding of function behavior.

Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given trigonometric expression. In the provided exercise, we simplified the equation to \( \cos^2 x \), making it easier to understand its graphical implications.
When tackling trigonometric equations, consider the following steps:

• Simplification: Use identities to transform the equation into a more tractable form. This might involve converting all functions into sine and cosine for simplicity.
• Graphical analysis: Graphing the simplified function helps in visualizing the possible solutions across different intervals.
• Finding solutions: Determine the specific values or intervals where solutions exist, considering the periodicity of trigonometric functions.

By mastering these steps, you can effectively solve complex trigonometric equations.
Consistent practice with equations like \( \cot^2 x - \cot^2 x \cos^2 x \) helps to solidify understanding and increases problem-solving skill level.