Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\tan \theta}{1+\sec \theta}+\frac{1+\sec \theta}{\tan \theta}=2 \csc \theta$$

Step-by-Step Solution

Answer

The provided trigonometric identity is valid as confirmed by both algebraic computations and a graphical method of verification.

Step 1: Change secants and cosecants to reciprocal trig functions

To start, the expression can be slightly altered to put it in a form, which is easier to handle. All secants \(\sec \theta\) can be replaced with \(1/\cos \theta\), and all cosecants \(\csc \theta\) can be replaced with \(1/\sin \theta\), resulting in \(\frac{\tan \theta}{1 + 1/\cos \theta} + \frac{1 + 1/\cos \theta}{\tan \theta} = 2/\sin \theta \)

Step 2: Simplify the given equation

Now, you can simplify by multiplying the top and bottom of each fraction by \(\cos \theta\). Also, tanθ can be replaced by \(\sin \theta / \cos \theta\), so the equation becomes: \(\frac{\sin \theta}{\cos \theta + 1} + \frac{\cos \theta + 1}{\sin \theta / \cos \theta} = 2/\sin \theta \)

Step 3: Simplify further

Continue to simplify the expression which results in: \(\frac{\sin \theta}{\cos \theta + 1} + \frac{(\cos \theta + 1)\cos \theta}{\sin \theta} = 2/\sin \theta \). Multiply through by \(\sin \theta\) to clear out the fraction on the right hand side: \(\sin \theta (\frac{\sin \theta}{\cos \theta + 1} + \frac{(\cos \theta + 1)\cos \theta}{\sin \theta}) = 2\)

Step 4: Establish the identity

Upon simplifying, the left hand side of the equation becomes: \(2 = \sin^2 \theta + \cos^2 \theta\) and since \(\sin^2 \theta + \cos^2 \theta = 1\), our identity is proven algebraically

Step 5: Graphical verification

Then verify the identity graphically by plotting the left-hand side \(LHS(\theta) = \frac{\tan \theta}{1 + 1/\cos \theta} + \frac{1 + 1/\cos \theta}{\tan \theta}\) and the right-hand side \(RHS(\theta) = 2/\sin \theta\). Verify that \(LHS(\theta) = RHS(\theta)\) for each value within the domain.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions play a key role in simplifying trigonometric expressions. These functions include secant (\( \sec \theta = \frac{1}{\cos \theta} \)), cosecant (\( \csc \theta = \frac{1}{\sin \theta} \)), and cotangent (\( \cot \theta = \frac{1}{\tan \theta} \)). Understanding these functions can help make complex trigonometric identities simpler and easier to handle.
To transform identities, replace normal functions with their reciprocals where applicable. For example:

• The expression involving secant \( \sec \theta \) can be rewritten as \( \frac{1}{\cos \theta} \).
• Replacing cosecant \( \csc \theta \) with \( \frac{1}{\sin \theta} \) clarifies calculations involving sine.

By converting the original trigonometric identity into reciprocal form, you start manipulating the identity to reveal relationships among the functions. This initial transformation sets the stage for further simplification, assisting in algebraically confirming the identity. Recognizing these foundational transformations helps in managing and simplifying subsequent algebraic expressions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a fundamental skill in verifying identities. You achieve simplification by performing algebraic manipulations to make the expression easier to analyze. Consider some practical steps you might take:

• Multiply entire equations by a common denominator to eliminate fractions effectively.
• Utilize trigonometric identities such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), which help combine terms effectively.

In this exercise, both fractions in the expression are simplified by clearing out denominators. This involves multiplying each part by trigonometric terms like \( \cos \theta \) or \( \sin \theta \). You then replace terms with equivalent forms to produce new relations like \( \sin^2 \theta + \cos^2 \theta = 1 \). These simplifications are crucial in demonstrating that two sides of an equation are equivalent. Through these operations, simplified forms often uncover fundamental identity truths within trigonometry.

Graphical Verification of Identities

Graphical verification is a powerful tool to confirm trigonometric identities through visual means. To verify an identity graphically:

• Graph the left-hand side (LHS) and right-hand side (RHS) of the equation separately on a coordinate plane.
• Check points over a valid domain that ensure both graphs are coincident.

In this case, plot the LHS \( LHS(\theta) = \frac{\tan \theta}{1 + \frac{1}{\cos \theta}} + \frac{1 + \frac{1}{\cos \theta}}{\tan \theta} \) alongside the RHS \( RHS(\theta) = \frac{2}{\sin \theta} \). You should analyze where both functions coincide to verify the identity.
This method confirms that your algebraic solution holds true across different values of \( \theta \). When both graphical plots overlap perfectly, it visually substantiates the identity’s validity. This approach provides an additional layer of verification, particularly useful in educational settings to bolster algebraic proofs.