Problem 63 - 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 since \( 1 - \sin^2 v = \cos^2 v \).

Step 1: Express in terms of sine and cosine

Start by expressing the terms \( \sec v \) and \( \tan v \) in terms of \( \sin v \) and \( \cos v \): \( \sec v = \frac{1}{\cos v} \) and \( \tan v = \frac{\sin v}{\cos v} \). This gives us: \[ \sec v - \tan v = \frac{1}{\cos v} - \frac{\sin v}{\cos v}. \]

Step 2: Combine the terms

Combine the expressions under a common denominator: \[ \sec v - \tan v = \frac{1 - \sin v}{\cos v}. \]

Step 3: Substitute and simplify right side

We aim for \( \frac{1}{\sec v + \tan v} \). Substitute \( \sec v = \frac{1}{\cos v} \) and \( \tan v = \frac{\sin v}{\cos v} \) into the right side: \[ \sec v + \tan v = \frac{1}{\cos v} + \frac{\sin v}{\cos v} = \frac{1 + \sin v}{\cos v}. \]

Step 4: Simplify right side to fractional form

The right side becomes: \[ \frac{1}{\sec v + \tan v} = \frac{1}{\frac{1 + \sin v}{\cos v}} = \frac{\cos v}{1 + \sin v}. \]

Step 5: Verify the identity

Now equate the expression for \( \sec v - \tan v = \frac{1 - \sin v}{\cos v} \) and \( \frac{1}{\sec v + \tan v} = \frac{\cos v}{1 + \sin v} \). By cross multiplication, we verify: \( (1 - \sin v)(1 + \sin v) = \cos^2 v \), which is an identity: \( 1 - \sin^2 v \equiv \cos^2 v \). Thus, the identity is verified.

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is one of the six main trigonometric functions. It is defined as the reciprocal of the cosine function. Mathematically, it is expressed as \( \sec \theta = \frac{1}{\cos \theta} \). This means that wherever cosine is zero, secant will be undefined because division by zero is not possible.

Secant is particularly useful in solving trigonometric identities and equations. It often appears in problems where relationships between angles must be understood in terms of their side ratios. Here's a key point to remember:

• \( \sec \theta \) is undefined for angles where \( \cos \theta = 0 \).
• \( \sec \theta \) increases as \( \cos \theta \) decreases and approaches zero.

In the context of verifying identities, expressing secant in terms of sine and cosine can simplify complex trigonometric equations. By converting secant to its cosine counterpart, it becomes easier to combine and manipulate expressions.

Tangent

The tangent function, symbolized as \( \tan \theta \), is defined as the ratio of the sine function to the cosine function for a given angle \( \theta \). In equation form, it is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function describes the slope of the angle and is fundamental in trigonometry because it relates the angles to the ratios of the sides of a right triangle.

Tangent is a periodic function with a period of \( \pi \, \text{radians} \) or 180 degrees. It plays a crucial role in defining other more complex trigonometric functions and in solving problems involving right angles:

• \( \tan \theta \) is undefined where \( \cos \theta = 0 \), as division by zero is not allowed.
• \( \tan \theta \) provides essential insights into the geometry of triangles.

In verification problems, rewriting tangent in terms of sine and cosine helps simplify equations, making it easier to identify and verify identities.

Verification of Trigonometric Identities

Verification of trigonometric identities is an essential skill because it allows us to prove that two different-looking expressions involving trigonometric functions are indeed equal. In other words, it's about using mathematical logic and known identities to show equivalence.

The process typically involves:

• Rewriting complex trigonometric expressions using basic identities like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Expressing all functions in terms of sine and cosine for consistency.
• Simplifying the expressions step-by-step until both sides of the equation match.

The goal is to transform the given expression using substitutions and algebraic manipulations until both sides of the identity appear identical. This kind of problem-solving is not only about calculations but also understanding relationships between trigonometric functions.

Cross Multiplication in Trigonometry

Cross multiplication is a method often used to eliminate fractions when verifying trigonometric identities. This technique involves multiplying across the diagonals of a fraction equation to facilitate simplification and manage complex expressions.

In the context of trigonometry, you might encounter this step when attempting to simplify different expressions to their simplest form. Here's how it works:

• Take each side of the equation and multiply it by the denominator of the opposite side.
• This process helps in transforming the equation into a simpler polynomial or quadratic form.

When verifying the identity \( \sec v - \tan v = \frac{1}{\sec v + \tan v} \), cross multiplication allows us to directly compare the polynomial expressions involved, leading to successful identity verification. Importantly, this method underpins understanding of why the two sides of an identity are equal and elucidates the connection between various trigonometric components.