Problem 35 - Technical Mathematics with Calculus

Question

Prove each identity. (All identities in this chapter can be proven. ) $$\left(\cos ^{2} \theta+\sin ^{2} \theta\right)^{2}=1$$

Step-by-Step Solution

Answer

\(1^{2} = 1\), therefore the identity is proven.

Step 1: Recognize the Pythagorean Identity

Recall that one of the fundamental Pythagorean trigonometric identities is \( \cos^2\theta + \sin^2\theta = 1 \). This identity states that the sum of the squares of sine and cosine of an angle equals to 1. In the given identity, \( (\cos ^{2} \theta + \sin ^{2} \theta)^{2} \), we see that the Pythagorean Identity is being squared.

Step 2: Apply the Pythagorean Identity

Since \( \cos^2\theta + \sin^2\theta = 1 \) holds for all angles \(\theta\), we can substitute 1 for the sum inside the square. Thus, we replace \( \cos^2\theta + \sin^2\theta \) with 1 in the given expression to get \(1^{2}\), which equates to \(1\).

Step 3: Square the Result

After substituting, we have \(1^{2}\), which is simply \(1\). Squaring 1 doesn't change its value because 1 multiplied by itself is still 1.

Key Concepts

Trigonometric Identity Proof

Understanding how to prove trigonometric identities is crucial in mastering the study of trigonometry. A trigonometric identity is an equation that is true for all values of the variables involved. Proving an identity involves showing that the left-hand side and right-hand side of the equation are equal for all values within the domain of the variables. The proof process can involve a range of techniques such as factoring, distributing, and using fundamental identities.

In our example, the identity to be proven is \(\left(\cos ^{2} \theta+\sin ^{2} \theta\right)^{2}=1\). This involves recognizing that the inside of the squared term on the left-hand side is a Pythagorean identity. By acknowledging that the Pythagorean identity, \(\cos^2\theta + \sin^2\theta = 1\), holds for any angle \(\theta\), it becomes evident that squaring a value which is always 1 will result in 1. As a result, the original expression simplifies directly to 1, providing a straightforward proof of the identity.

Sine and Cosine Relationship

A fundamental concept in trigonometry is the relationship between sine and cosine. These functions are connected not only visually through the unit circle but also algebraically through identities. The Pythagorean theorem, when applied to the unit circle, leads to the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\), which shows a direct relationship between the sine and cosine of an angle. This equation tells us that for any angle \(\theta\), the square of the cosine added to the square of the sine will always equal 1.

Understanding this relationship is essential when working with various trigonometric expressions and when solving for unknown sides in right-angled triangles. This relationship is so critical it can be used to define one function in terms of the other; for example, \(\sin^2\theta = 1 - \cos^2\theta\) and vice versa.

Mathematical Identities

The term mathematical identities refers to equations that hold true for all allowed values of the variables they contain. These identities are an integral part of problem-solving in various areas of mathematics, including algebra, calculus, and especially trigonometry. Mathematical identities provide a way to simplify complex expressions and to compare different forms of expressions to discern if they are equivalent.

An important distinction must be made between an identity and an equation: while an equation is true for specific values, an identity is a universal truth. The understanding and application of mathematical identities not only aid in proving higher-level concepts but also foster a deeper comprehension of fundamental mathematical operations and connections.