Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x=\csc ^{2} x$$

Step-by-Step Solution

Answer

Yes, the identity is verified: \( \csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x=\csc ^{2} x\)

Step 1: Break down the expressions

First, break down the expressions such that they are represented in terms of sin and cos. This gives: \(\frac{1}{\sin x}(\frac{1}{\sin x}-\sin x)+\frac{\sin x-\cos x}{\sin x}+\frac{\cos x}{\sin x}\).

Step 2: Simplify the expression

Simplify the above expression by combining terms with the same denominator, which is \(\sin x\). This will result into: \(\frac{1- \sin^2 x + \sin x -\cos x + \cos x}{\sin x}\).

Step 3: Use Pythagorean identity

Utilize the Pythagorean identity, \(1- \sin^2 x = \cos^2 x\), the expression becomes: \frac{\cos^2 x + \sin x}{\sin x}.

Step 4: Further Simplification

Rearrange the terms and this expression simplifies to: \(\frac{\cos^2 x}{\sin x} + 1\), which is equal to: \( \csc ^{2} x\), which verifies the original identity.

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room. We take a complex expression and simplify it to make it easier to understand. By transforming parts of the expression into simpler forms or common fractions, we reveal the essence of the problem.

In the exercise provided, our goal was to break down the complex expression \( \csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x \) into simpler terms. We started by rewriting the trigonometric functions in terms of \( \sin x \) and \( \cos x \):

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

From there, every part of the expression was expressed as a fraction with a common denominator, \( \sin x \), simplifying the addition of fractions.

This process resulted in the expression: \[ \frac{1- \sin^2 x + \sin x -\cos x + \cos x}{\sin x} \] Learning to simplify like this is powerful because it reveals relationships between mathematical elements that might not be immediately obvious.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry, often used to simplify expressions. It is based on the famous Pythagorean theorem and shows the relation between the sine and cosine of an angle. The identity is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

In the original exercise, this identity was crucial in rearranging and simplifying the expression further.

We utilized the rearranged form:

• \( 1- \sin^2 x = \cos^2 x \)

Applying this, we transformed part of the complex expression: \[ \frac{1- \sin^2 x + \sin x}{\sin x} \to \frac{\cos^2 x + \sin x}{\sin x} \] Understanding and applying such trigonometric identities allows us to tackle more complex trigonometric problems with confidence and ease.

Graphical Verification

Graphical verification involves using a graph to explore and confirm mathematical identities visually. This approach complements algebraic solutions by offering a visual understanding of the problem and its solution.

A graphing utility or tool can be used to plot both sides of the equation individually. For our exercise, you would graph \( \csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x \) and \( \csc^2 x \) as separate functions. By observing their plots, if they coincide over the domain of interest, it confirms the algebraic identity.

• This provides a second layer of verification ensuring accuracy.
• Visual tools help identify discrepancies or showcase the equivalence of equations through the overlap of graphs.

This method of confirmation is especially helpful for visual learners who benefit from seeing concepts expressed graphically. It also enhances the understanding of the function's behavior across different values of \( x \).