Problem 98 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \((1+\sin x+\cos x)^{2}=2(1+\sin x)(1+\cos x)\)

Step-by-Step Solution

Answer

The equation is proven to be an identity.

Step 1: Expand the Left Side

Let's first expand the left side of the equation, \[(1+\sin x+\cos x)^2.\] Using the formula for the square of a trinomial, \\( (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \), we get: \[ 1^2 + (\sin x)^2 + (\cos x)^2 + 2(1)(\sin x) + 2(\sin x)(\cos x) + 2(\cos x)(1). \]This simplifies to: \[ 1 + \sin^2 x + \cos^2 x + 2\sin x + 2\sin x \cos x + 2\cos x. \] Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), this further simplifies to: \[ 2 + 2\sin x + 2\cos x + 2\sin x \cos x. \]

Step 2: Expand the Right Side

Now, expand the right side of the equation: \[ 2(1+\sin x)(1+\cos x). \] Using the formula for the product of two binomials, \((a+b)(c+d) = ac + ad + bc + bd \), we find:\[ 2(1 \cdot 1 + 1 \cdot \cos x + \sin x \cdot 1 + \sin x \cdot \cos x). \] Simplifying inside the parentheses first gives:\[ 2(1 + \cos x + \sin x + \sin x \cos x). \] Distributing the 2 yields:\[ 2 + 2\cos x + 2\sin x + 2\sin x \cos x. \]

Step 3: Compare the Results

Now compare the results from both sides:* Left side: \[ 2 + 2\sin x + 2\cos x + 2\sin x \cos x \]* Right side: \[ 2 + 2\sin x + 2\cos x + 2\sin x \cos x \]Both the expressions are identical, confirming that the original equation is indeed an identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that, for any angle \(x\), the sum of the square of sine and cosine is always equal to 1. This is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is very useful because it allows us to simplify trigonometric expressions by converting between sine and cosine. For example, in the original exercise, using the Pythagorean identity simplifies the expansion:

• If \( \sin^2 x + \cos^2 x = 1 \), then you can replace this sum in any equation with 1.

This simplification often helps to verify identities, solve equations, or even prove other trigonometric formulas. Recognizing when and how to use this identity is a key skill in trigonometry.

Binomial Theorem

The binomial theorem provides a way to expand expressions that are raised to a power. It applies directly to terms of the form \((a + b)^n\). For the case where \(n\) is 2, which we use in the original exercise, the expression is expanded as follows:

• \( (a+b)^2 = a^2 + 2ab + b^2 \)

Often used in algebra and trigonometry, this theorem helps break down more complex expressions into manageable parts. In our exercise, we expanded trinomials such as \((1 + \sin x + \cos x)^2\) using a similar expansion formula. This approach allows for a straightforward calculation of each term, making complex identities more digestible. Strengthening this understanding will enhance your ability to work efficiently with polynomials and other algebraic operations.

Expansion of Expressions

Expansion of expressions involves transforming a compact expression into an extended form that shows each component of the expression clearly. This technique is crucial when working with mathematical expressions, as it allows for precise manipulation and simplification. For the problem at hand, expanding both sides of the identity:

• Left side: Expand \((1 + \sin x + \cos x)^{2}\) using the formula for the square of a trinomial.
• Right side: Expand \(2(1 + \sin x)(1 + \cos x)\) using the formula for a product of two binomials.

Once expanded, each individual term is clear and can be evaluated separately or in groups. This clarity is vital in comparing complex expressions like we did in the original exercise. Expansion effectively breaks down potentially overwhelming algebraic expressions into smaller, well-defined components, making it accessible to confirm identities or solve equations.