Problem 92 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sin ^{3} x-\cos ^{3} x}{\sin x-\cos x}=1+\sin x \cos x\)

Step-by-Step Solution

Answer

The given trigonometric identity \(\frac{\sin ^{3} x-\cos ^{3} x}{\sin x-\cos x}=1+\sin x \cos x\) is verified to be correct.

Step 1: Recognize and apply the formula

Looking at the numerator in the given identity, it is recognized that it's in the form of \(a^3 - b^3\). So, we apply the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = \sin x\) and \(b = \cos x\). This lets us rewrite the left side of the equation as: \(\frac{(\sin x-\cos x)(\sin^2 x + \sin x \cos x + \cos^2 x)}{\sin x-\cos x}\)

Step 2: Simplify the terms

We simplify the left side of the equation by canceling out \(\sin x - \cos x\) from the numerator and denominator. This results in: \(\sin^2 x + \sin x \cos x + \cos^2 x\).

Step 3: Apply the Pythagorean identity

Next step is to apply the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\). By substituting this in the simplified equation from the previous step, we get: \(1 + \sin x \cos x\). Which is what we have on the right side of the equation. Hence, the identity is verified.

Key Concepts

Verifying Trigonometric Identities

The process of verifying trigonometric identities involves proving that two sides of an equation are equivalent for all values of the variable that make each expression defined. The fundamental approach includes using known identities, like those of Pythagorean, reciprocal, or quotient identities, to manipulate the expressions and demonstrate their equality.

It is essential to familiarize yourself with these basic identities as they form the building blocks for verifying more complicated expressions. When verifying identities, you should aim to transform one side of the equation to exactly match the other or show that both sides simplify to a common expression. Patience and practice are key, since this process often involves several steps of algebraic manipulation.

Pythagorean Identity

The Pythagorean identity is a fundamental equality in trigonometry, which relates the squares of sine and cosine functions. It states that for any angle, the sum of the square of sine and the square of cosine equals one: \( \(sin^2 x + cos^2 x = 1\) \).

This identity is derived from the Pythagorean Theorem, a principle in geometry that deals with the relations of sides in a right triangle. The Pythagorean identity is an invaluable tool in simplifying trigonometric expressions and is frequently used in the process of verifying trigonometric identities, as it allows the conversion of one trigonometric function into another.

Algebraic Manipulation

Algebraic manipulation encompasses the techniques used to rearrange and simplify algebraic expressions and equations, which is vital in solving and verifying trigonometric identities. This set of skills includes operations such as factoring, expanding, adding, subtracting, multiplying, and dividing expressions, as well as canceling common factors.

In the context of trigonometric identities, algebraic manipulation often involves recognizing patterns that resemble formulas or identities, as was done with the cubic difference in the exercise. Understanding the properties of equations and inequalities can also guide how we manipulate an expression to maintain equivalence between both sides of an identity. Mastering algebraic manipulation makes it much easier to verify trig identities and is a necessary proficiency in higher mathematics.