Problem 63 - Precalculus: Functions and Graphs

Question

Either show that the equation \(i s\) an identity or show that the equation \(is\quad not\) an identity. $$\cos x(\tan x+\cot x)=\csc x$$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Analyze the left-hand side (LHS)

The expression on the LHS is \( \cos x (\tan x + \cot x) \). We need to simplify this expression to see if it equals the right-hand side (RHS), which is \( \csc x \). Let's start by simplifying \( \tan x + \cot x \).

Step 2: Simplify \(\tan x + \cot x\)

Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Therefore, \( \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \). Find a common denominator to combine these fractions: \( \tan x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} \).

Step 3: Use Pythagorean Identity

Remember that \( \sin^2 x + \cos^2 x = 1 \) by the Pythagorean identity. So, the expression becomes \( \frac{1}{\sin x \cos x} \).

Step 4: Combine with \(\cos x\) from LHS

Substitute the simplified form of \( \tan x + \cot x \) back into the LHS expression, resulting in \( \cos x \cdot \frac{1}{\sin x \cos x} \). This simplifies further to \( \frac{1}{\sin x} \), which is the definition of \( \csc x \).

Step 5: Compare with RHS

The simplified LHS \( \frac{1}{\sin x} = \csc x \) matches exactly with the RHS, \( \csc x \). Therefore, the equation is an identity.

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry. It's a crucial tool used in many mathematical proofs and simplifications. The most well-known Pythagorean identity is \( \sin^2 x + \cos^2 x = 1 \). This formula is derived from the Pythagorean theorem applied to the unit circle.

Whenever you see \( \sin^2 x + \cos^2 x \) in an expression, you can substitute it with 1 to simplify the expression. This identity works because in a right triangle inscribed in a unit circle, the sum of the squares of the sine and cosine (the two other sides of the right triangle) must be equal to the square of the hypotenuse, which is always 1 in the unit circle.

This identity is not only essential for simplifying expressions such as those involving tangent and cotangent functions, but it also plays a foundational role in calculus and other areas of mathematics. Recognizing the Pythagorean identity helps you transform complex trigonometric expressions into simpler forms, unlocking solutions in a straightforward manner.

trigonometric functions

Trigonometric functions are the building blocks of trigonometry, a branch of mathematics that studies the relationships between the sides and angles of triangles. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function relates to a specific ratio of sides in a right triangle, depending on the angle you're examining.

• \( \sin x \) is the ratio of the opposite side to the hypotenuse.
• \( \cos x \) is the ratio of the adjacent side to the hypotenuse.
• \( \tan x \) is the ratio of the opposite side to the adjacent side and can also be expressed as \( \frac{\sin x}{\cos x} \).

Besides these, there are reciprocal trigonometric functions: cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). These are the reciprocals of sine, cosine, and tangent, respectively. Understanding how these functions interact and relate to one another is key when working with trigonometric identities and solving complex equations.

In trigonometric identities, knowing these reciprocal relationships helps in transforming and simplifying expressions, often turning challenging problems into more manageable ones.

simplifying trigonometric expressions

Simplifying trigonometric expressions involves using known identities and algebraic manipulation to make expressions easier to work with or compare. It's a vital skill in solving trigonometric equations, proving identities, or preparing equations for limits and integrals in calculus.

To simplify trigonometric expressions, one often uses identities such as Pythagorean, reciprocal, and quotient identities. It's important to recognize patterns like \( \sin^2 x + \cos^2 x \) or to transform complex fractions using common denominators, as in the exercise problem. For instance, in simplifying \( \tan x + \cot x \), recognizing their equivalent forms \( \frac{\sin x}{\cos x} \) and \( \frac{\cos x}{\sin x} \) allows us to combine them into a single fraction with a common denominator.

Once you've simplified the fractions, the use of reciprocal identities can transform expressions further. In the problem, simplifying \( \frac{1}{\sin x \cos x} \) and then multiplying it by \( \cos x \) eventually reveals the simple form \( \csc x \). This shows the power of simplifying trigonometric expressions where complex-looking equations can boil down to recognizable trigonometric functions.