Problem 40 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\sin ^{3} t+\cos ^{3} t=(1-\sin t \cos t)(\sin t+\cos t)$$

Step-by-Step Solution

Answer

The identity is verified, both sides are equal: \[(sin t+cos t)(1-sin t \cos t)\]

Step 1: Recollect the Basic Formula

We need to verify the identity:\[sin^3 t + cos^3 t = (1 - sin t \cos t)(sin t + cos t)\]To do this, recall the formula for sum of cubes: \[a^3 + b^3 = (a+b)(a^2 - ab + b^2)\] for any two real numbers \(a\) and \(b\).

Step 2: Apply the Sum of Cubes Formula

Let \(a = sin t\) and \(b = cos t\). Then:\[sin^3 t + cos^3 t = (sin t + cos t)(sin^2 t - sin t \cdot cos t + cos^2 t)\]Since \(sin^2 t + cos^2 t = 1\), the expression simplifies to:\[ (sin t + cos t)(1 - sin t \cdot cos t)\]

Step 3: Simplify the Right-Hand Side

The right side of the identity is:\[(1 - sin t \cos t)(sin t + cos t)\]This is already in the same form as the expression we simplified in Step 2: \[(sin t + cos t)(1 - sin t \cos t)\]

Step 4: Compare Both Sides

The simplified left-hand side expression is equal to the given right-hand side expression:\[(sin t + cos t)(1 - sin t \cos t)\]This shows that both sides of the equation are the same, verifying the identity.

Key Concepts

Sum of Cubes

The sum of cubes formula is an essential mathematical identity that helps simplify the addition of two cubed terms. It is expressed as:

• For any real numbers \(a\) and \(b\), the sum of their cubes can be written as follows:\[a^3 + b^3 = (a+b)(a^2 - ab + b^2)\]

This formula simplifies expressions involving cubes and is a mathematically concise way to understand cubic combinations.
In the context of the given exercise, \(a\) is equal to \(\sin t\) and \(b\) is equal to \(\cos t\). Applying the sum of cubes formula helps transform the original identity on the left side from \(\sin^3 t + \cos^3 t\) into:

• \((\sin t + \cos t)(\sin^2 t - \sin t \cos t + \cos^2 t)\)

This transposition reveals a more tractable form and plays a crucial role in verifying that both sides of the equation are, indeed, equivalent.

Trigonometric Simplification

Trigonometric simplification involves using known identities and properties to make trigonometric expressions easier to understand or solve. This technique is applied to reduce expressions using identities or combine them into a simpler form.
In the provided problem, trigonometric simplification involves substituting and restructuring terms. Once we rewrote the left side of the identity using the sum of cubes, a known trigonometric property helps simplify it further.

• By recognizing that \(\sin^2 t + \cos^2 t = 1\), one of the Pythagorean Identities frequently used in trigonometric simplification, we replace \(\sin^2 t + \cos^2 t\) with 1.
• This step effectively simplifies the expression inside the parentheses to:\[(\sin t + \cos t)(1 - \sin t \cos t)\]

This shows how leveraging basic trigonometric identities can reduce a complicated expression to a recognizable identity, making calculations easier and verifying the original identity simple.

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental and widely used trigonometric identities. It is based on the Pythagorean Theorem from geometry and is a critical tool in simplifying trigonometric expressions and proving trigonometric identities.

• The basic Pythagorean Identity states:\[\sin^2 t + \cos^2 t = 1\]
• This identity is true for any angle \(t\) and is derived from the Pythagorean Theorem applied to the unit circle where the radius is 1.

In our exercise, the Pythagorean Identity is used to simplify the cube sum:
When rewriting \(\sin^3 t + \cos^3 t\), we incorporated this identity by replacing \(\sin^2 t + \cos^2 t\) with 1, which simplified the expression to match the form of the right-hand side:

• \((\sin t + \cos t)(1 - \sin t \cos t)\)

The correct recognition and usage of this identity not only verify the given trigonometric identity but also underscore the importance of foundational trigonometric principles in simplifying expressions effectively.