Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cot \alpha}{\csc \alpha-1}=\frac{\csc \alpha+1}{\cot \alpha}$$

Step-by-Step Solution

Answer

By taking cotangent and cosecant in terms of sine and cosine, and further simplifying, and applying Pythagorean identity and simplifying again, it is shown that the two sides of the given equation are identical. Graphically, you would see that the plots of the two expressions coincide completely, confirming the algebraic verification done previously.

Step 1: Express in terms of sine and cosine

Express cotangent and cosecant in terms of sine and cosine. Therefore, the expression becomes: \(\frac{\frac{1}{\tan \alpha}}{\frac{1}{\sin \alpha}-1}=\frac{\frac{1}{\sin \alpha}+1}{\frac{1}{\tan \alpha}}\)

Step 2: Simplify the equation

Simplify the equation by eliminating the fractions within the fractions. This gives us: \(\frac{\cos \alpha}{\sin \alpha(1-\sin \alpha)} = \frac{1+\sin \alpha}{\cos \alpha}\)

Step 3: Equate the expressions

Equate these two expressions to verify the identity: \(\frac{\cos \alpha}{\sin \alpha-\sin^2 \alpha} = \frac{1+\sin \alpha}{\cos \alpha}\)

Step 4: Cross Multiply

Cross multiply the equation to verify the identity : \(\cos^2 \alpha(1+\sin \alpha) = \cos \alpha(\sin \alpha-\sin^2 \alpha)\)

Step 5: Simplify further

Simplify the equation further and apply Pythagorean identity : \(1-\sin^2 \alpha + \cos^2 \alpha \sin \alpha =\cos \alpha \sin \alpha -\cos \alpha \sin^2 \alpha\)

Step 6: Simplify to Verify Identity

Cancel out terms on both sides to verify the equation.

Step 7: Final step: Graphing

For graphical verification, plot both the expressions to see if they coincide.

Key Concepts

Trigonometric Functions

Trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each function is related to the angles in a right triangle, and they describe the relationship between the angles and side lengths of the triangle.
To work with trigonometric identities, it often helps to express these functions in terms of sine and cosine. For example:

• Cotangent: \(\cot \alpha = \frac{1}{\tan \alpha} = \frac{\cos \alpha}{\sin \alpha}\)
• Cosecant: \(\csc \alpha = \frac{1}{\sin \alpha}\)

These conversions simplify identities and equations, making the algebraic manipulations more straightforward. By consistently expressing other trigonometric functions in terms of these two, resolving equations becomes almost like a puzzle, just with several solvable pieces.

Algebraic Manipulation

Algebraic manipulation is an essential skill when proving trigonometric identities. It involves rearranging, factoring, and simplifying equations to show that two expressions are equivalent.
In problems like this, eliminate complex fractions within fractions. For example, turning expressions like \(\frac{\frac{1}{\tan \alpha}}{\frac{1}{\sin \alpha} - 1}\) into something more manageable helps a lot.
This manipulation includes:

• Cancelling Out: Resolve complex fractions by multiplying numerators and denominators to eliminate extra divisions.
• Simplifying: Combine like terms, reduce complex terms to simpler forms, and look for factors that both sides share.
• Cross Multiplication: Use this technique to eliminate fractions by multiplying across the expression, ensuring each side of the equation maintains equivalence.

These steps refine the equation, making it easier to identify if the identity holds through further simplifications.

Graphical Verification

Graphical verification is a powerful method to visually confirm whether a trigonometric identity holds. This approach uses graphing tools to plot the two expressions you want to verify as identical.
When both expressions are plotted and appear as the same curve on the graph, it confirms they are equivalent. This step is crucial for verifying identities beyond algebraic methods, showing how an equation behaves over a range of angles.
To perform this:

• Use a graphing calculator or software to input both expressions \(\frac{\cot \alpha}{\csc \alpha - 1}\) and \(\frac{\csc \alpha + 1}{\cot \alpha}\).
• Plot each expression separately over a defined interval for \(\alpha\).
• Check that both plots coincide with each other entirely.

This visual method complements algebraic checks, providing a more comprehensive assurance of the identity's validity.

Pythagorean Identity

The Pythagorean identity is one of the fundamental relations in trigonometry. It connects the square of sine and cosine to one. Formally, it is expressed as:\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
This identity helps simplify complex trigonometric equations by substituting for \(\sin^2 \alpha\) or \(\cos^2 \alpha\).
Using this identity in algebraic manipulation provides a direct path to simplifying expressions like those seen in the verification process.

• While simplifying, if you encounter parts like \(1 - \sin^2 \alpha\), substitute it with \(\cos^2 \alpha\).
• Likewise for expressions involving \(1 - \cos^2 \alpha\) use \(\sin^2 \alpha\).

These substitutions simplify the equation and help verify identities by bringing them into a form more readily compared. This fundamental identity is pivotal in the process of demonstrating the equivalence of two trigonometric expressions.