Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\frac{\cos x-\cos y}{\sin x+\sin y}+\frac{\sin x-\sin y}{\cos x+\cos y}=0$$

Step-by-Step Solution

Answer

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Step 1: Add the two fractions

To start solving this, notice that we can add the two fractions by finding a common denominator. The common denominator of \(\sin x+\sin y\) and \(\cos x+\cos y\) is \((\sin x+\sin y)(\cos x+\cos y)\). Therefore, we obtain \(\frac{(\cos x-\cos y)(\cos x+\cos y)+(\sin x-\sin y)(\sin x+ \sin y)}{(\sin x+ \sin y)(\cos x+ \cos y)}\).

Step 2: Apply Difference of Squares

For the numerator of the fraction: Notice that \((\cos x - \cos y)(\cos x + \cos y)\) and \((\sin x - \sin y)(\sin x + \sin y)\) are in the form of a difference of squares, \(a^{2} - b^{2}\) which can be factored into \((a-b)(a+b)\). Applying this, we get \(\frac{(\cos^{2}x-\cos^{2}y)+(\sin^{2}x- \sin^{2}y)}{(\sin x + \sin y)(\cos x+ \cos y)}\).

Step 3: Use Pythagorean Identity

Next, use the Pythagorean identity \(\sin^{2}x + \cos^{2}x = 1\) and \(\sin^{2}y + \cos^{2}y = 1\), allowing to further simplify the numerator, \(\frac{(1-\cos^{2}y)+(1- \sin^{2}y)}{(\sin x + \sin y)(\cos x+ \cos y)} = \frac{2}{(\sin x+ \sin y)(\cos x+ \cos y)}\).

Key Concepts

Graphing Utility

A graphing utility is a valuable tool in mathematics that helps us visualize equations and functions. It can plot points, draw graphs, and, most importantly, create tables of values.
These tables make it easy to check solutions numerically by evaluating the function at different points.
Using the table feature in a graphing utility allows you to input various values for the variables involved in an equation. This can be particularly helpful when verifying trigonometric identities.
To verify an identity like the one in our problem, you would input specific values for \(x\) and \(y\), then calculate the left-hand side and right-hand side of the equation.
If they match for different pairs of values, it's a good indicator that the identity may be true. However, a formal proof like we have shown isn't replaced by numerical evidence – because numerical evidence can be coincidental.

Difference of Squares

The difference of squares is a mathematical pattern that applies to expressions of the form \(a^2 - b^2\).
This pattern is particularly useful in our trigonometric identity problem, where it was used to simplify the equation.The formula works as \((a - b)(a + b) = a^2 - b^2\).
This means if you see expressions multiplied like \((\cos x - \cos y)(\cos x + \cos y)\), they can be rewritten as \(\cos^2 x - \cos^2 y\).
The same principle applies to the sine terms too.Using the difference of squares simplifies the expression to something manageable by eliminating the middle terms of the expansion, leaving a nice, straightforward expression in terms of squares, which can then be easily manipulated with trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry that relates the sine and cosine functions.
The most common form of it is \(\sin^2 x + \cos^2 x = 1\).
This identity is derived from the Pythagorean theorem applied to the unit circle.When you work with trigonometric identities, this relationship often allows for the replacement or simplification of terms.
For instance, in problems like ours, you can simplify expressions like \(1 - \cos^2 y\) to \(\sin^2 y\), and vice versa.
Using this identity transformed our complex fraction into a more recognizable and simple form, leading us to confirm the given identity.The Pythagorean identity is essential in simplifying and solving numerous trigonometric problems effectively, making it a powerful tool in any mathematics toolkit.