Problem 47 - Contemporary Precalculus

Question

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t+\cos t)(\sin t-\cos t)$$

Step-by-Step Solution

Answer

Answer: \((\sin t + \cos t)(\sin t - \cos t) = 1 - 2\cos^2 t\)

Step 1: Identify the correct trigonometric identity

To simplify the given expression, we should recognize that it has the form of a difference of squares, which can be written as \((a + b)(a - b) = a^2 - b^2\). In this case, \(a = \sin t\) and \(b = \cos t\).

Step 2: Apply the difference of squares identity

Now we substitute the expressions for \(a\) and \(b\) into the difference of squares formula: \((\sin t + \cos t)(\sin t - \cos t) = (\sin t)^2 - (\cos t)^2\)

Step 3: Use the Pythagorean trigonometric identity

Recall that the Pythagorean identity for sine and cosine functions is \(\sin^2 t + \cos^2 t = 1\). We can use this identity to simplify our expression further by expressing one of the trigonometric functions in terms of the other one. In this case, we'll substitute \((1 - \cos^2 t)\) for \(\sin^2 t\). \((\sin t)^2 - (\cos t)^2 = (1 - \cos^2 t) - (\cos t)^2\)

Step 4: Simplify the expression

Now, combine the terms in the expression: \((1 - \cos^2 t) - (\cos t)^2 = 1 - 2\cos^2 t\) The simplified expression is:

Step 5: Answer:

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

Key Concepts

Difference of Squares

When you encounter a problem involving the product of two binomials that are identical except for the opposite signs, you're dealing with a difference of squares. Specifically, the formula is \(a^2 - b^2 = (a + b)(a - b)\). Understanding this identity is essential in algebra since it simplifies what might initially appear like a complex expression. By recognizing this structure in equations and expressions, you can swiftly simplify them by performing the multiplication to get a binomial where the middle terms cancel out, leaving you with the difference of the squares of the two terms.

Pythagorean Identity

The Pythagorean identity is another vital concept in trigonometry. It states that for any angle \(t\), \(\sin^2 t + \cos^2 t = 1\). This identity is derived from the Pythagorean theorem relative to the unit circle, where the sine and cosine represent the lengths of the sides of a right triangle. If you can rewrite one trigonometric function in terms of the other using this identity, it significantly simplifies the process of solving trigonometric expressions.

Simplifying Expressions

Simplifying algebraic expressions makes them easier to work with. By applying the rules of arithmetic and algebra, you can reduce expressions to a more manageable form. This often involves combining like terms, factoring, using identities (such as the Pythagorean identity), and canceling terms. Simplification helps not only in getting the answer but also in understanding the structure of the problem and the relationships between the mathematical components within it.

Algebraic Manipulation

Algebraic manipulation involves actions like expanding, factoring, simplifying, and solving, using various algebraic rules and properties. It is the toolkit that students use to shape algebraic expressions to their simplest form or in a form that is most usable for a particular purpose. Mastery of these techniques allows students to tackle a broad range of mathematical problems, including those with complicated trigonometric identities, by transforming them into more familiar and solvable forms.