Problem 77 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x $$

Step-by-Step Solution

Answer

The identity is verified; both sides simplify to \( 2 \sec x \).

Step 1: Simplify the First Fraction

We start with the first fraction \( \frac{1}{\sec x + \tan x} \). Multiply the numerator and the denominator by \( \sec x - \tan x \) to rationalize it. This gives:\[\frac{\sec x - \tan x}{(\sec x + \tan x)(\sec x - \tan x)} = \frac{\sec x - \tan x}{\sec^2 x - \tan^2 x}\]Use the identity \( \sec^2 x - \tan^2 x = 1 \) to simplify to:\[\sec x - \tan x.\]

Step 2: Simplify the Second Fraction

Now take the second fraction \( \frac{1}{\sec x - \tan x} \). Multiply the numerator and the denominator by \( \sec x + \tan x \) to rationalize it. This results in:\[\frac{\sec x + \tan x}{(\sec x - \tan x)(\sec x + \tan x)} = \frac{\sec x + \tan x}{\sec^2 x - \tan^2 x}\]Since \( \sec^2 x - \tan^2 x = 1 \), the expression simplifies to:\[\sec x + \tan x.\]

Step 3: Add the Two Simplified Expressions

Add the results from Step 1 and Step 2:\[(\sec x - \tan x) + (\sec x + \tan x) = 2 \sec x\]Here, the \( - \tan x \) and \( + \tan x \) cancel each other, leaving us with a single term:\[2 \sec x.\]

Step 4: Compare with the Original Equation

The resulting sum from Step 3, \( 2 \sec x \), matches the right side of the identity:\[\frac{1}{\sec x + \tan x} + \frac{1}{\sec x - \tan x} \equiv 2 \sec x.\]Hence, the identity is verified.

Key Concepts

Secant Function

The secant function, represented as \( \sec x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. In mathematical terms, \( \sec x = \frac{1}{\cos x} \).
This function is particularly useful when working with reciprocal identities in trigonometry.
Unlike cosine, which is the ratio of the adjacent side to the hypotenuse in a right triangle, secant is the ratio of the hypotenuse to the adjacent side. Here's why secant is important:

• It gives us insight into how the angle \( x \) relates to lengths in a right triangle.
• Understanding secant helps in solving problems involving reciprocal identities and simplifying equations involving cosines.
• The secant function tends to infinity as \( x \) approaches \( \pi/2 + k\pi \) for integers \( k \), which can have significant implications in calculus or complex trigonometric simplifications.

Tangent Function

The tangent function, denoted as \( \tan x \), is another primary trigonometric function. It is the ratio of the sine function to the cosine function. Thus, the definition is \( \tan x = \frac{\sin x}{\cos x} \).
Tangent provides a way to relate the opposite and adjacent sides of a right triangle with the angle \( x \). It's important to note that:

• Tangent experiences asymptotes (undefined points) whenever \( \cos x = 0 \), which occurs at \( x = \pi/2 + k\pi \).
• Tangent simplifies many trigonometric identities, especially when combined with other functions like secant and cotangent.

In practical terms, knowing how tangent behaves enables us to handle more complex trigonometric expressions effectively, as seen in trigonometric simplifications and solving equations.

Rationalizing Denominators

Rationalizing the denominator is a technique used in algebra and trigonometry to eliminate irrational or complex numbers from the denominator of a fraction.
To rationalize a denominator, one typically multiplies both the numerator and the denominator by a conjugate of the denominator or other suitable expression.
The process ensures that the denominator becomes a rational number or simpler expression. Here are some key points:

• For trigonometric expressions like \( \frac{1}{\sec x + \tan x} \), multiplying by a conjugate such as \( \sec x - \tan x \) helps simplify the denominators.
• This method transforms the denominator using identities like \( \sec^2 x - \tan^2 x = 1 \), which arises from the Pythagorean identity \( \sec^2 x = 1 + \tan^2 x \).
• Rationalizing helps streamline complex expressions, making them easier to combine or reduce further.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions in simpler or more familiar forms, often by using identities.
The main goal is to make expressions more manageable for evaluation, solving equations, or integration. Some essential aspects include:

• Using identities like \( \sec^2 x - \tan^2 x = 1 \), which help simplify fractions involving trigonometric functions.
• Rationalizing and adding expressions, as demonstrated in the exercise, leads to simplifications like \( 2 \sec x \).
• By reducing expressions using identities and algebraic manipulation, one can attest to the integrity of trigonometric equations and gain deeper insights.

Trigonometric simplification plays a crucial role in verifying identities, solving trigonometric equations, and proving theorems in mathematics.