Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the multiplication and use the fundamental identities to simplify. $$(5-5 \sin x)(5+5 \sin x)$$

Step-by-Step Solution

Answer

The simplification for the given multiplication using the fundamental identities results in \(25 \cos^2 x\)

Step 1: Recognize the pattern

We can recognize that this is a multiplication of two binomials that follows the pattern \((a-b)(a+b) = a^2 - b^2\). In this case, \(a = 5\) and \(b = 5 \sin x\).

Step 2: Apply the formula

Apply the formula \(a^2 - b^2\). We get \(25 - (5 \sin x)^2 = 25 - 25\sin^2 x\).

Step 3: Use fundamental trigonometric identity

The fundamental trigonometric identity is \(\sin^2 x + cos^2 x = 1\). From this, we can rewrite \(\sin^2 x\) as \(1 - \cos^2 x\). So, the expression becomes \(25 - 25(1 - \cos^2 x)\).

Step 4: Simplify the equation

After simplification, we get \(25 - 25 + 25 \cos^2 x = 25 \cos^2 x\).

Key Concepts

Binomial Multiplication

When you see an expression like \((5 - 5 \sin x)(5 + 5 \sin x)\), it might look complicated at first. It's important to recognize that this is a form of binomial multiplication. Binomials are expressions with two terms, and multiplying them follows a straightforward pattern.

Here, the pattern is known as the difference of squares. The formula \((a-b)(a+b) = a^2 - b^2\) simplifies our task significantly. In this particular example, \(a\) is 5, and \(b\) is \(5 \sin x\).

Thus, the expression becomes \(a^2 - b^2 = 25 - (5 \sin x)^2\). By applying this pattern, you've reduced a multiplication problem into a simpler subtraction problem. Recognizing and using such patterns are crucial skills in simplifying expressions.

Simplifying Trigonometric Expressions

Trigonometric expressions often look complex, but with practice, they can become fairly straightforward to work with. Once you've applied the binomial multiplication, you end up with the expression \(25 - 25\sin^2 x\). The goal now is to simplify it further.

Simplification often involves recognizing and using known identities or rearranging terms to get rid of unnecessary complexity.

• First, notice that each term in the expression has a common factor, \(25\), which allows you to factor the expression as follows: \(25(1 - \sin^2 x)\).
  
• This small step of factoring can make the simplification easier and the expression cleaner.
  

Through simplification, you transform a complex equation into a more manageable one.

Pythagorean Identity

One of the most useful identities in trigonometry is the Pythagorean Identity: \(\sin^2 x + \cos^2 x = 1\). This identity is fundamental because it allows us to express \(\sin^2 x\) in terms of \(\cos^2 x\) and vice versa.

In the context of our problem, this identity plays a key role. After performing binomial multiplication and initial simplification, we reach the expression \(25 - 25\sin^2 x\). Using the identity, we can replace \(\sin^2 x\) with \(1 - \cos^2 x\).

Thus, the expression simplifies to:

• Begin with: \(25 - 25(\sin^2 x)\)
• Substitute \(\sin^2 x = 1 - \cos^2 x\)
• Simplify the expression: \(25 - 25 + 25\cos^2 x\)
• Finally, this yields: \(25\cos^2 x\)

With such substitutions, complex trigonometric expressions become much easier to handle, showcasing the power of trigonometric identities.