Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 61-64: Either show that the equation is an identity or show that the equation is not an identity. $$ (\sec x+\tan x)^{2}=2 \tan x(\tan x+\sec x) $$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Expand Left Side of Equation

Start with the left side of the equation \((\sec x + \tan x)^2\). This can be expanded using the formula for a square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\) to give \[\sec^2 x + 2 \sec x \tan x + \tan^2 x.\]

Step 2: Simplify Using Trigonometric Identities

Use the identity \(\sec^2 x = 1 + \tan^2 x\) to further simplify terms. Replace \(\sec^2 x\) in the expanded form: \[1 + \tan^2 x + 2 \sec x \tan x + \tan^2 x = 1 + 2\tan^2 x + 2 \sec x \tan x.\]

Step 3: Expand Right Side of Equation

The right side of the equation is \(2 \tan x(\tan x + \sec x)\). Distribute \(2 \tan x\) into the parenthesis: \[2 \tan x \times \tan x + 2 \tan x \times \sec x = 2 \tan^2 x + 2 \sec x \tan x.\]

Step 4: Compare Both Sides

Notice that both sides of the equation are now the same. Left side: \[1 + 2 \tan^2 x + 2 \sec x \tan x.\] Right side: \[2 \tan^2 x + 2 \sec x \tan x.\] Since \(1 + 2 \tan^2 x + 2 \sec x \tan x = 2 \tan^2 x + 2 \sec x \tan x + 1\), both sides are equal.

Step 5: Conclusion

Since both sides of the equation are identical after simplification, the given equation is an identity. This means it holds true for all values of \(x\) where the functions are defined.

Key Concepts

Trigonometric Equations

A trigonometric equation is one that involves trigonometric functions like sine, cosine, tangent, etc. It often requires finding values that satisfy the equation across a range of angles. These equations can be equations with identities, where both sides are equal for all values of the variable within the domain of the functions involved. In the example exercise, we are dealing with an equation that was identified as an identity because both sides of the equation simplify to the same expression for any value of the variable where the trigonometric functions are defined. The key to solving such problems lies in skillfully using trigonometric identities and algebraic manipulation. By strategically expanding and simplifying both sides of the equation, we verify if both sides match, which indicates an identity. Understanding these equations is crucial since they are foundational in calculus and physics, where trigonometric models describe wave patterns, oscillations, and other phenomena.

Algebraic Manipulation

Algebraic manipulation is a method to transform and work with mathematical expressions to simplify or solve them. It's akin to playing with building blocks—rearranging and changing the components to see the relationships more clearly. In the exercise, algebraic manipulation was used to expand and simplify both sides of the given trigonometric equation.

• Starting with the left side, applying the expansion of a binomial square equation: \((a+b)^2 = a^2 + 2ab + b^2\).
• The implementation of a known identity \(\sec^2 x = 1 + \tan^2 x\).
• Similarly, the right side was expanded by distributing \(2 \tan x \).

The comparisons showed that both expressions were equivalent, thus verifying the equation's identity. Becoming proficient at these manipulations involves practicing the distribution, combining like terms, and applying identities correctly.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry and relate angles of a triangle to the lengths of its sides. They include sine, cosine, tangent, cosecant, secant, and cotangent. Each function represents a ratio, such as tangent \(\tan(x) = \frac{\sin x}{\cos x}\) or secant \(\sec(x) = \frac{1}{\cos x}\). In this exercise, we worked with tangent and secant functions.

• The secant represents the reciprocal of cosine, while tangent represents the ratio of sine to cosine.
• Trigonometric identities, such as \(\sec^2 x = 1 + \tan^2 x\), allow for simplifying expressions involving these functions.

These identities are crucial when solving equations or transforming complex trigonometric expressions. Understanding how these functions interplay through identities helps in problem-solving and understanding more advanced mathematical topics.