Problem 85 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x $$

Step-by-Step Solution

Answer

The identity is verified; both sides simplify to \( 1 - \sin x \cos x \).

Step 1: Understand the Identity Structure

Identify the structure of the expression we need to verify. The given identity is \( \frac{\sin ^{3} x + \cos ^{3} x}{\sin x + \cos x} = 1 - \sin x \cos x \). We are required to show that the left-hand side (LHS) equals the right-hand side (RHS).

Step 2: Factorize the Numerator

The expression \( \sin^3 x + \cos^3 x \) can be factorized using the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, \( a = \sin x \) and \( b = \cos x \), so:\[\sin ^{3} x + \cos ^{3} x = (\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x)\]

Step 3: Simplify the Expression

Substitute the factorized form of the numerator into the fraction:\[\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = \frac{(\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x)}{\sin x + \cos x}\]Since \( \sin x + \cos x \) appears in both the numerator and denominator, it cancels out, leaving:\[\sin^2 x - \sin x \cos x + \cos^2 x\]

Step 4: Use Pythagorean Identity

Recognize that \( \sin^2 x + \cos^2 x = 1 \) from the Pythagorean identity. Substitute this into the simplified expression:\[\sin^2 x + \cos^2 x = 1 \Rightarrow \sin^2 x - \sin x \cos x + \cos^2 x = 1 - \sin x \cos x\]

Step 5: Verify the Equality

We have found that both the left-hand side and right-hand side of the original equation simplify to \( 1 - \sin x \cos x \). Therefore, the given identity is verified.

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is an algebraic identity used for factorizing expressions of the type \( a^3 + b^3 \). It can be tricky at first, but with a bit of practice, you'll see how it transforms mathematical expressions and helps simplify complex algebra. The formula itself is:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

When applying this to trigonometric expressions such as \( \sin^3 x + \cos^3 x \), consider:

• Set \( a = \sin x \) and \( b = \cos x \).

You replace \( a^3 + b^3 \) with \( (a + b)(a^2 - ab + b^2) \), and this can greatly aid in breaking down the expression into a more manageable form.
It's a powerful tool that helps in reducing trinomial expressions, especially when working with identities. Apply this whenever you encounter similar cube expressions for a smoother algebraic process.

Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry, helping in simplifying expressions involving sine and cosine functions. Stemming from the Pythagorean theorem, this identity states:

• \( \sin^2 x + \cos^2 x = 1 \)

By itself, the identity summarizes how the square of the sine added to the square of the cosine will always equal one for any angle \( x \).
This might seem trivial, but it is immensely useful in verifying and simplifying trigonometric expressions. For example, in our exercise, we used this identity in step 4 to transition from the expression \( \sin^2 x - \sin x \cos x + \cos^2 x \) to \( 1 - \sin x \cos x \).
Whenever you see \( \sin^2 x + \cos^2 x \) in an expression, it's always a good approach to convert it to 1 using this identity. It's one of those must-know tools in your trigonometry toolkit.

Trigonometric Simplification

Trigonometric simplification involves steps that reduce complex expressions to their simplest forms. To successfully carry out this task, a solid understanding of trigonometric identities is necessary. Simplifications, like the one in our exercise, often revolve around canceling terms and substituting equal values.

• Identify any common factors in the numerator and denominator, such as \( \sin x + \cos x \) in our example.
• Utilize basic trigonometric identities to substitute equivalent expressions.

In the given problem, canceling \( \sin x + \cos x \) from both the numerator and the denominator was key to simplifying the expression to the form \( \sin^2 x - \sin x \cos x + \cos^2 x \), allowing further simplification using the Pythagorean identity.
By applying these steps methodically, troublesome trigonometric equations become much clearer and more approachable. Always handle each component of the equation carefully, maintaining balance and integrity throughout the simplification process.