Problem 35 - Precalculus with Limits

Question

In Exercises 29-36, use a double-angle formula to rewrite the expression. \( (\cos x + \sin x)(\cos x - \sin x) \)

Step-by-Step Solution

Answer

The expression \( (\cos x + \sin x)(\cos x - \sin x) \) can be rewritten using a double-angle formula as \( \cos{2x} \)

Step 1: Apply the difference of two squares formula

This formula states that for any two numbers \( a \) and \( b \), their difference multiplied by their sum (i.e., \( (a + b)(a - b) \)) can be simplified to \( a^2 - b^2 \). Apply this to given expression to achieve \( \cos^{2}x - \sin^{2}x \).

Step 2: Double-angle formula application

At this point, the double-angle formula can be used to further simplify the expression. The double-angle identity for cosine, \( \cos{2u} \) is given by \( \cos^{2}u - \sin^{2}u \). Substituting \( u \) with \( x \), the given expression becomes \( \cos{2x} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's particularly famous for its functions like sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. These functions are fundamental in solving problems involving right-angled triangles, and they also extend their utility to non-right angled situations through the laws of sines and cosines.

The vast spectrum of trigonometry covers not only basic angle calculations but also complex applications in fields such as physics, engineering, and even astronomy. Understanding trigonometry is essential for students who require both theoretical and practical knowledge to solve various geometric and physics problems.

Difference of Two Squares

Difference of two squares is an algebraic pattern that occurs frequently in mathematics and has various applications, including in trigonometry. The formula states that if you have two terms, let's call them 'a' and 'b', the product of the sum and the difference of these two terms (\( a + b \) and \( a - b \) ) will always equal the difference between the square of the first term and the square of the second, which can be represented as \( a^2 - b^2 \) .

This is a key concept because it allows us to simplify expressions and solve equations that might otherwise appear complex. It's particularly handy in trigonometry because many trigonometric identities involve squared terms, and this formula can often be used to simplify them.

Double-Angle Identity

The double-angle identity is a fundamental concept in trigonometry that shows the relationship between the cosine or sine of twice an angle and the squares of the cosine or sine of the original angle. These identities come in handy when solving trigonometric equations and can greatly simplify complex expressions.

For cosine, the double-angle formula can be expressed in three different ways:

• \( \text{cos}(2x) = \text{cos}^2(x) - \text{sin}^2(x) \)
• \( \text{cos}(2x) = 2\text{cos}^2(x) -1 \)
• \( \text{cos}(2x) = 1 - 2\text{sin}^2(x) \)

This versatility is particularly useful because you can choose the most convenient form depending on the given problem or the expression you are trying to simplify.

Cosine

In trigonometry, the cosine function, abbreviated as 'cos', is one of the primary trigonometric functions. It is related to an angle of a right triangle, defining the ratio of the adjacent side length to the hypotenuse. For an angle \( \theta \) , the cosine function is represented as \( \text{cos}(\theta) \) .

The cosine function is also pivotal in the study of waves, oscillations, and many other areas of mathematics and physics. Its value ranges between -1 and 1. The fundamental properties of the cosine function, such as its period, amplitude, and phase shift, are essential for understanding and solving problems related to periodic phenomena.