Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: 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, as both sides are equal after simplification.

Step 1: Expand the Right-Hand Side (RHS)

Let's begin by expanding the right-hand side of the equation \((1 - \sin t \cos t)(\sin t + \cos t)\). Use the distributive property to expand:\[(1 - \sin t \cos t)(\sin t + \cos t) = 1 \cdot \sin t + 1 \cdot \cos t - \sin t \cos t \cdot \sin t - \sin t \cos t \cdot \cos t\]This simplifies to:\[\sin t + \cos t - \sin^2 t \cos t - \sin t \cos^2 t\]

Step 2: Simplify the Terms

Further simplify the expression from Step 1:\[\sin t + \cos t - \sin^2 t \cos t - \sin t \cos^2 t = \sin t + \cos t - \sin t \cos t (\sin t + \cos t)\]Factor out the common term \(\sin t \cos t\):\[\sin t + \cos t = (\sin t + \cos t) - \sin t \cos t (\sin t + \cos t)\]

Step 3: Check for Identity of LHS with Simplified RHS

The left-hand side of the equation is \(\sin^3 t + \cos^3 t\). We use the identity for sum of cubes:\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]Applying this identity, we get:\[\sin^3 t + \cos^3 t = (\sin t + \cos t)(\sin^2 t - \sin t \cos t + \cos^2 t)\]Since \(\sin^2 t + \cos^2 t = 1\), we have:\[\sin^2 t - \sin t \cos t + \cos^2 t = 1 - \sin t \cos t\]This gives us:\[\sin^3 t + \cos^3 t = (\sin t + \cos t)(1 - \sin t \cos t)\]

Step 4: Final Verification

We have shown that both sides of the original equation simplify to:\[(\sin t + \cos t)(1 - \sin t \cos t) = (1 - \sin t \cos t)(\sin t + \cos t)\]Thus, the identity is verified.

Key Concepts

Sum of Cubes

When we talk about the sum of cubes, we're referring to an algebraic identity that is useful in simplifying expressions. The standard identity is: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] This means that if you add two cubes together, you can factor them using this identity.
In trigonometry, we often apply this identity to expressions involving trigonometric functions like \( \sin^3 t + \cos^3 t \). Here, \( a \) and \( b \) are \( \sin t \) and \( \cos t \), respectively. By using this formula, we simplify the expression into a more manageable form.

• The expression becomes \((\sin t + \cos t)(\sin^2 t - \sin t \cos t + \cos^2 t)\).
• It's useful in identities verification and simplifies complex trigonometric expressions.

Distributive Property

The distributive property is a fundamental rule in algebra, which helps when multiplying a single term by two or more terms within parentheses. It states that: \[a(b + c) = ab + ac\] This allows us to expand expressions, making it a valuable tool when working with trigonometric identities.
In our specific context, we applied it to the expression \((1 - \sin t \cos t)(\sin t + \cos t)\) to expand and simplify:

• First, distribute \(1\) to both \( \sin t \) and \( \cos t \).
• Then, distribute \(-\sin t \cos t\) to both \( \sin t \) and \( \cos t \).

As a result, we simplify down to a more usable expression. This process is crucial to solving many algebraic and trigonometric problems efficiently.

Simplifying Expressions

Simplifying expressions is all about reducing them to their simplest form, making them easier to work with. This involves combining like terms, using identities, and removing unnecessary components.
In the task of verifying trigonometric identities, simplifying is a crucial step:

• After expansion with the distributive property, we gathered similar terms, like \( \sin t\) and \( \cos t \).
• We factored out common terms where possible - in this case, \( \sin t \cos t \).

This leads to a simpler, equivalent expression which aligns with the other side of the equation, demonstrating the veracity of the given identity.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are essential tools in mathematics that help describe the relationships between the angles and sides of triangles. They also appear frequently in identities used across various applications.
Sine \( (\sin t) \) and cosine \( (\cos t) \) are especially common. They relate to the coordinates of a point on a unit circle.

• Quadratic identity: \( \sin^2 t + \cos^2 t = 1 \).
• This identity is utilized in simplifying expressions, as seen in \( \sin^2 t - \sin t \cos t + \cos^2 t = 1 - \sin t \cos t \).

By using these functions and identities, complex expressions become more tractable, emphasizing their usefulness in problem solving and verification.