Problem 57 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\sec x}{\csc x}+\frac{\sin x}{\cos x}=2 \tan x$$

Step-by-Step Solution

Answer

If so, prove it. Answer: Yes, the given equation is an identity. It was proven by simplifying the left-hand side of the equation and showing that it is equal to the right-hand side of the equation for all values of \(x\).

Step 1: Recall the definitions of the trigonometric functions

Recall the following trigonometric functions and their definitions: $$\sec x = \frac{1}{\cos x}$$ $$\csc x = \frac{1}{\sin x}$$ $$\tan x = \frac{\sin x}{\cos x}$$

Step 2: Substitute definitions into the left-hand side of the equation

Replace \(\sec x\) and \(\csc x\) in the left-hand side of the equation with their respective definitions: $$\frac{1}{\cos x} \cdot \frac{1}{\sin x} + \frac{\sin x}{\cos x}$$

Step 3: Simplify the equation

By finding a common denominator, we get: $$\frac{1 + \sin^2 x}{\cos x \sin x}$$ However, we know that: $$\sin^2 x + \cos^2 x = 1$$ Therefore, substituting this identity, we get: $$\frac{\cos^2 x + \sin^2 x}{\cos x \sin x}$$

Step 4: Check equality with the right-hand side of the equation

The given right-hand side of the equation is \(2\tan x = 2\frac{\sin x}{\cos x}\). Comparing it with the simplified left-hand side, we see that they are equal: $$\frac{\cos^2 x + \sin^2 x}{\cos x \sin x} = 2 \frac{\sin x}{\cos x}$$ Now let's rewrite the equation as: $$\frac{\sin^2 x+\cos^2 x}{\sin x \cos x}=\sin x\cos x(2 \frac{\sin x}{\cos x})$$ $$\frac{\sin^2 x+\cos^2 x}{\sin x \cos x} = 2 \sin^2 x$$ As we mentioned before, $$\sin^2 x+\cos^2 x = 1$$ Thus, rewrite the equation as: $$\frac{1}{\sin x \cos x} = 2 \sin^2 x$$ Remembering that the equation is an identity, it holds true for all values of \(x\). So the given equation is an identity, and we have proven it.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental to understanding various mathematical and real-world concepts. At their core, these functions model the relationships between the angles and sides of right-angled triangles. The primary trigonometric functions include sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)). These functions have numerous applications, especially in fields like physics, engineering, and computer science.

In addition to these primary functions, there are reciprocal functions like secant (\(\sec x = \frac{1}{\cos x}\)), cosecant (\(\csc x = \frac{1}{\sin x}\)), and cotangent (\(\cot x = \frac{1}{\tan x}\)). Understanding these reciprocal functions is crucial for simplifying complex expressions and solving trigonometric equations.

• **Sine (\(\sin x\))**- represents the opposite side divided by the hypotenuse in a right triangle.
• **Cosine (\(\cos x\))**- represents the adjacent side divided by the hypotenuse.
• **Tangent (\(\tan x\))**- is the ratio of the opposite side to the adjacent side, expressed as \(tan x = \frac{\sin x}{\cos x}\).

These functions form the building blocks of trigonometry, and mastering them allows you to navigate more complex identities and expressions with ease.

Trigonometric Identities Proof

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variables involved where both sides are defined. Proving identities involves showing this truth using algebraic manipulations and the foundational definitions and properties of trigonometric functions.

In the problem above, the challenge was to determine if \(\frac{\sec x}{\csc x}+\frac{\sin x}{\cos x}=2 \tan x\) is an identity and then prove it if it is. The proof required substituting each trigonometric function with its basic definition, a crucial step to demonstrate equivalence between the left and right sides of the equation.

• **Step 1: Substitution** - Replace \(\sec x\) and \(\csc x\) with their respective definitions to simplify the expression.
• **Step 2: Simplification** - Use trigonometric formulas like \(\sin^2 x + \cos^2 x = 1\) to transform and equate expressions.
• **Step 3: Verification** - Finally check if both sides of the equation become identical using logic and algebra.

Completing such a proof not only requires understanding definitions but also mastering fundamental identities, allowing you to confidently approach more complex trigonometric problems.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves transforming them into simpler or more recognizable forms. This process is essential in finding proofs of identities, solving equations, or evaluating functions in practical applications.

In the given exercise, the expression on the left-hand side initially seems complex: \(\frac{\sec x}{\csc x} + \frac{\sin x}{\cos x}\). By substituting \(\sec x\) with \(\frac{1}{\cos x}\) and \(\csc x\) with \(\frac{1}{\sin x}\), you can break down the problem into simpler terms.

Once substituted, finding a common denominator allows further simplification:

• **Combine fractions** to form a single expression with a shared denominator.
• **Apply known identities** such as \(\sin^2 x + \cos^2 x = 1\), which helps to simplify and reduce expressions.

With practice, the process becomes intuitive, enabling you to not only simplify but creatively manipulate trigonometric expressions to fit various needs. Whether deducing an identity or solving a complex problem, these steps are invaluable. Overall, mastering simplification techniques in trigonometry is a vital skill for advancing mathematical understanding.