Problem 63 - 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. $$ \cos x(\tan x+\cot x)=\csc x $$

Step-by-Step Solution

Answer

The equation is an identity as both sides are equal for all defined values of \( x \).

Step 1: Understanding the Equation

The given equation is \( \cos x(\tan x + \cot x) = \csc x \). We need to verify if both sides are equal for all values of \( x \) where both expressions are defined.

Step 2: Simplify Left-Hand Side (LHS)

The left-hand side of the equation is \( \cos x(\tan x + \cot x) \). Substitute the identities \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \), which gives \( \cos x\left( \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \right) \). This simplifies to \( \cos x\left( \frac{\sin^2 x + \cos^2 x}{\cos x \sin x} \right) \).

Step 3: Utilize Pythagorean Identity

Now, use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Substitute this into the equation to get \( \cos x\left( \frac{1}{\cos x \sin x} \right) = \frac{1}{\sin x} \).

Step 4: Simplify Further

The expression \( \cos x \times \frac{1}{\cos x \sin x} \) simplifies to \( \frac{1}{\sin x} = \csc x \).

Step 5: Compare Both Sides of Equation

Now we have \( \frac{1}{\sin x} = \csc x \). Thus, the left-hand side equals the right-hand side of the equation for all defined values, indicating that the original equation is an identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states that for any angle \( x \), the identity \( \sin^2 x + \cos^2 x = 1 \) always holds true. This equation is derived from the Pythagorean theorem, hence its name.
Understanding and applying this identity is crucial when simplifying trigonometric expressions. It essentially ties together the sine and cosine functions in a way that can simplify otherwise complex calculations.

• For example, if you know the value of \( \sin x \), this identity allows you to easily find \( \cos x \).
• This identity can be transformed for different uses, such as expressing \( \sin^2 x \) in terms of \( \cos^2 x \), like so: \( \sin^2 x = 1 - \cos^2 x \).

In our specific exercise, the Pythagorean identity is used to simplify the expression \( \sin^2 x + \cos^2 x \) into \( 1 \), which plays a critical role in verifying that the given equation is indeed an identity.

Trigonometric Simplification

Trigonometric simplification involves reducing a complex expression into a simpler or more familiar form using trigonometric identities and algebraic manipulation.
This step is essential for solving or verifying equations involving trigonometric functions.
To simplify an expression, you may:

• Use basic trigonometric identities, like \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).
• Look for opportunities to apply identities such as the Pythagorean identity, reciprocal identities (e.g., \( \csc x = \frac{1}{\sin x} \)), and even double angle formulas, depending on the problem.

In this particular exercise, the left-hand side \( \cos x(\tan x + \cot x) \) is simplified using basic identities which transform it into a well-known trigonometric term, \( \csc x \). This successful simplification validates the equation as an identity through straightforward algebra along with the application of the Pythagorean identity.

Proof of Identity

Proving an identity in trigonometry requires demonstrating that two expressions are equivalent for all angles for which they are defined. This involves mathematical manipulations, substitutions with trigonometric identities, and simplifications.
The proof may include the following steps:

• Identify both sides of the equation and consider starting with the more complex side to simplify it.
• Use relevant identities to rewrite terms into a common or simplified form.
• Systematically simplify or alter the expression until both sides match.

In our exercise, we verified the identity by simplifying the left-hand side using substitutions for \( \tan x \) and \( \cot x \), followed by employing the Pythagorean identity. As a result, both sides equated to \( \csc x \), proving that the original equation holds true universally for all valid angles. Each step builds logically upon the previous ones, confirming the identity effectively and systematically.