Problem 64 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x$$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sec x+\csc x}{\tan x+\cot x}=\sin x+\cos x \).

Step 1: Express reciprocal trigonometric functions in terms of sine and cosine

The first step is to express the reciprocal trigonometric functions in the given equation using sine and cosine. We know that \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), \( \tan x = \frac{\sin x}{\cos x} \), and \( \cot x = \frac{\cos x}{\sin x} \). Substitute these into the equation: \[ \frac{\frac{1}{\cos x} + \frac{1}{\sin x}}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} = \sin x + \cos x \]

Step 2: Simplify the numerator of the left-hand side

The numerator on the left-hand side is \( \frac{1}{\cos x} + \frac{1}{\sin x} \). To combine these fractions, find a common denominator, which is \( \cos x \sin x \), and rewrite the expression:\[ \frac{\sin x + \cos x}{\cos x \sin x} \]

Step 3: Simplify the denominator of the left-hand side

The denominator is \( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \). Again, find a common denominator, which is \( \cos x \sin x \), and simplify:\[ \frac{\sin^2 x + \cos^2 x}{\cos x \sin x} \]Using the identity \( \sin^2 x + \cos^2 x = 1 \), the expression becomes:\[ \frac{1}{\cos x \sin x} \]

Step 4: Divide the simplified numerator by the simplified denominator

Now, divide the simplified numerator by the simplified denominator:\[ \frac{\sin x + \cos x}{\cos x \sin x} \div \frac{1}{\cos x \sin x} = (\sin x + \cos x) \times \cos x \sin x = \sin x + \cos x \]

Step 5: Verify identity

Both sides of the equation, after simplification, are equal:\[ \sin x + \cos x = \sin x + \cos x \]This confirms that the given identity is verified.

Key Concepts

Reciprocal trigonometric functions

Reciprocal trigonometric functions are essentially the flips or inverses of the basic trigonometric functions like sine, cosine, and tangent. They have specific names: secant, cosecant, and cotangent. Understanding these functions requires a firm grasp on how they relate to the primary functions:

• Secant (\( \sec x \)): This is the reciprocal of the cosine function. The formula is given by \( \sec x = \frac{1}{\cos x} \), meaning it takes the reciprocal of the cosine ratio.
• Cosecant (\( \csc x \)): This is the reciprocal of sine. Its formula is \( \csc x = \frac{1}{\sin x} \), once again representing its inverse nature.
• Cotangent (\( \cot x \)): This is the reciprocal of the tangent function. It's expressed as \( \cot x = \frac{1}{\tan x} \) or equivalently \( \frac{\cos x}{\sin x} \).

For exercises involving trigonometric identities, substituting these functions with their equivalent sine and cosine forms is a crucial step. The conversions make it easier to simplify expressions and solve trigonometric equations efficiently.

Simplification of fractions

Simplification of fractions in trigonometric identities often involves combining terms under a common denominator. This simplifies expressions, which is essential for solving or verifying identities. Let's break it down:

• Finding a common denominator: This is the first step to combine fractions. Multiplying different fractions’ denominators gives a common ground to add or subtract them. For instance, with \( \frac{1}{\cos x} + \frac{1}{\sin x} \), the common denominator is \( \cos x \sin x \).
• Rewriting the numerators: Once you have a common denominator, adjust the numerators to reflect that new denominator: \( \frac{\sin x + \cos x}{\cos x \sin x} \) represents the unified form of the previous fractions.
• Simplifying complex fractions: Perform algebraic operations necessary to reduce fractions to their simplest form. This might involve using trigonometric identities such as \( \sin^2 x + \cos^2 x = 1 \), as they help simplify the expression to basic terms easily understood in verification problems.

Mastering these steps aids greatly in reducing the complexity of trigonometric expressions and makes verifying identities an attainable task.

Verifying identities

Verifying trigonometric identities involves showing that two different expressions are equivalent. This is a common type of problem in trigonometry, requiring clear understanding and strategic simplification. Here’s how you can approach it:

• Simplification: Start by simplifying both sides of the given identity separately. Use algebra and known trigonometric identities to make expressions more manageable.
• Substitution: Replace complex terms using basic trigonometric definitions and identities to see if simpler forms emerge. The identity \( \sin^2 x + \cos^2 x = 1 \) is particularly useful for substitution as seen in simplifying the equation to \( \frac{1}{\cos x \sin x} \).
• Equating expressions: With both sides reduced, check if their simplified forms are identical, proving the identity's validity. In this problem, both sides of the equation reduce to \( \sin x + \cos x \), showing they are indeed equal.

This methodical verification of identities is crucial for strengthening understanding of trigonometric concepts and improves problem-solving skills, promoting confidence in tackling even the most intricate problems.