Problem 52 - Algebra and Trigonometry

Question

Verify each identity. \(\sin ^{4} t-\cos ^{4} t=1-2 \cos ^{2} t\)

Step-by-Step Solution

Answer

Using the Pythagorean identity and difference of squares formula, the given trigonometric identity \( \sin ^{4} t-\cos ^{4} t=1-2 \cos ^{2} t \) has been verified.

Step 1: Analyze the given equation

The given equation is \( \sin ^{4} t-\cos ^{4} t=1-2 \cos ^{2} t \) . We can break down this equation into more manageable parts by using the difference of squares formula \(a^2 - b^2 = (a+b)(a-b)\), which is applicable because \( \sin^2 t \) and \( \cos^2 t \) are essentially squares.

Step 2: Apply the Difference of Squares

Apply the formula on the left side to get \( (\sin^2 t - \cos^2 t)(\sin^2 t + \cos^2 t) \). We know that \( \sin^2 t + \cos^2 t = 1 \) (basic Pythagorean identity). This simplifies the equation to \( (\sin^2 t - \cos^2 t) \times 1 \).

Step 3: Further Simplify the Equation

The formula \( \sin^2 t = 1 - \cos^2 t \) can now be used to replace \( \sin^2 t \) in the equation, which gives \( 1 - 2 \cos^2 t = 1 - 2 \cos^2 t \). After simplification, the left side of the equation becomes equal to the right side, thereby verifying the identity.

Key Concepts

Difference of Squares

The difference of squares formula is a handy tool in algebra, and it's extremely useful in trigonometry as well. This formula states that any expression in the form of \( a^2 - b^2 \) can be rewritten as \((a+b)(a-b)\). Why is this important? Well, it allows us to break down complex expressions into simpler parts, making them easier to work with. In the context of trigonometry, if you have \( \sin^4 t - \cos^4 t \), you can view it as \( (\sin^2 t)^2 - (\cos^2 t)^2 \), which perfectly fits the difference of squares pattern.

Here’s how you apply it:

• Identify the squared terms \( \sin^2 t \) and \( \cos^2 t \).
• Transform the expression using \((\sin^2 t + \cos^2 t)(\sin^2 t - \cos^2 t)\).
• Simplify using known identities, which makes your life a whole lot easier!

Breaking down expressions using this approach is not only elegant but also highly efficient in solving and verifying trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. It states that \( \sin^2 t + \cos^2 t = 1 \). This simple yet powerful identity is derived from the Pythagorean Theorem and is fundamental in simplifying and transforming trigonometric expressions. When verifying or simplifying expressions, knowing this identity can help you replace complex terms with simpler equivalents.

Whenever you see \( \sin^2 t \) or \( \cos^2 t \) in an equation, you might remember:

• \( \sin^2 t = 1 - \cos^2 t \)
• \( \cos^2 t = 1 - \sin^2 t \)

These transformations are invaluable for breaking down and simplifying equations, especially when verifying identities like \( \sin^4 t - \cos^4 t = 1 - 2 \cos^2 t \). With these tools, the solution process becomes more intuitive and streamlined.

Simplification Techniques

Simplification is often the most crucial step in verifying trigonometric identities. It involves using a variety of strategies to make an expression easier to handle. The goal is to transform a complicated equation into a simpler, equivalent form that is easy to verify.

In the context of the given problem, we start by using the difference of squares to decompose the expression. By applying the Pythagorean identity, the expression simplifies further. But there’s more to simplification:

• Recognize common factors and patterns.
• Substitute equivalent expressions using identities.
• Keep expressions balanced—what you do to one side, apply to the other.

All these steps help us confirm that the given identity holds true. With practice, these techniques become second nature, making the process of working with trigonometric identities much more approachable and less daunting.