Problem 65 - Algebra and Trigonometry

Question

Verify the identity. $$ \sec v-\tan v=\frac{1}{\sec v+\tan v} $$

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Use Trigonometric Definitions

The identity involves secant and tangent functions. We need to express them in terms of sine and cosine. We know:\[ \sec v = \frac{1}{\cos v} \] \[ \tan v = \frac{\sin v}{\cos v} \] Substitute these definitions into the identity.

Step 2: Substitute Definitions into the Identity

Replace \( \sec v \) and \( \tan v \) in the equation:\[ \frac{1}{\cos v} - \frac{\sin v}{\cos v} = \frac{1}{\frac{1}{\cos v} + \frac{\sin v}{\cos v}} \] This simplifies to:\[ \frac{1 - \sin v}{\cos v} = \frac{1}{\frac{1 + \sin v}{\cos v}} \]

Step 3: Simplify Right Side of the Equation

Focus on the right side of the equation:\[ \frac{1}{\frac{1 + \sin v}{\cos v}} = \frac{\cos v}{1 + \sin v} \]

Step 4: Confirm the Equation

Now compare both sides of the equation:Left: \( \frac{1 - \sin v}{\cos v} \)Right: \( \frac{\cos v}{1 + \sin v} \)To show they are equal:Multiply numerator and denominator of the left side by \(1 + \sin v\):\[ \frac{(1 - \sin v)(1 + \sin v)}{\cos v (1 + \sin v)} = \frac{1 - \sin^2 v}{\cos v (1 + \sin v)} \]Since \(1 - \sin^2 v = \cos^2 v\), this simplifies to:\[ \frac{\cos^2 v}{\cos v (1 + \sin v)} = \frac{\cos v}{1 + \sin v} \]Both sides are equal.

Key Concepts

Secant Function

In trigonometry, the secant function, denoted as \( \sec \), is one of the fundamental functions and is the reciprocal of the cosine function. It is defined as:

• \( \sec v = \frac{1}{\cos v} \)

This identity is useful in various trigonometric simplifications, especially when dealing with complex expressions involving multiple trigonometric functions. In the original problem, we used the definition of secant to transform the expression \( \sec v - \tan v \), making it easier to work with by expressing everything in terms of sine and cosine.

Understanding that \( \sec \) is the reciprocal of \( \cos \) helps in transforming complex fractions and simplifying many trigonometric equations. When working on a verification like this, always start by rewriting all functions as their sine or cosine equivalents whenever possible. This is because sine and cosine are the building blocks of all other trigonometric functions.

Tangent Function

The tangent function, represented by \( \tan \), is another crucial trigonometric function. It is expressed in terms of sine and cosine as follows:

• \( \tan v = \frac{\sin v}{\cos v} \)

This particular relationship plays a key role in simplifying trigonometric identities, such as the one given in the exercise. By substituting \( \tan v \) with \( \frac{\sin v}{\cos v} \), it's easier to handle expressions since they are all under a common denominator.

This method is especially useful when you need to manipulate or verify complex identities. By reducing the number of different trigonometric terms in an expression, it becomes easier to combine terms and simplify the overall expression. This form also aligns tangent with the idea of slope in a right triangle, offering a geometric interpretation that is both intuitive and helpful in solving problems.

Trigonometric Simplification

Trigonometric simplification involves modifying mathematical expressions to make them more manageable. In this exercise, simplification started by expressing secant and tangent functions in terms of sine and cosine. The key steps were:

• Substituting definitions like \( \sec v = \frac{1}{\cos v} \) and \( \tan v = \frac{\sin v}{\cos v} \).
• Transforming the given equation \( \sec v - \tan v = \frac{1}{\sec v + \tan v} \) into a single denominator form to make equations look similar.
• Using algebraic identities like \( 1 - \sin^2 v = \cos^2 v \) to match both sides of the equation.

This simplification process revealed the path to arrive at the identity's truth. By simplifying each side of the equation to match a common form, we confirm that an identity holds true. This skilful manipulation often relies on recognizing and applying fundamental trigonometric identities, such as Pythagorean identities, in clever ways to manipulate and simplify expressions efficiently.