Problem 43 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \cos^2 \beta + \cos^2 \left(\dfrac{\pi}{2} - \beta \right) = 1 \)

Step-by-Step Solution

Answer

Yes, the given identity \( \cos^2 \beta + \cos^2 \left(\frac{\pi}{2} - \beta \right) = 1 \) is correct as it successfully verifies the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \) where \( \theta = \beta \).

Step 1: Change of Variables

Replace \( \frac{\pi}{2} - \beta \) by \( \alpha \). So \( \alpha + \beta = \frac{\pi}{2} \). This helps to visualize that the two angles are complementary.

Step 2: Express Cosine in terms of Sine

Use the identity \( \cos \alpha = \sin \left(\frac{\pi}{2} - \alpha \right) \) to write \( \cos^2 \left(\frac{\pi}{2} - \beta \right) \) as \( \sin^2 \beta \). So our equation becomes \( \cos^2 \beta + \sin^2 \beta \).

Step 3: Use Pythagorean identity

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) where \( \theta = \beta \). Substitute \( \beta \) to get the final equation \( \cos^2 \beta + \sin^2 \beta = 1 \), which verifies the given identity.

Key Concepts

Cosine Squared

When developing a firm grasp of trigonometry, understanding the 'cosine squared' function, denoted as \( \cos^2(x) \), is crucial. This function represents the square of the cosine of an angle 'x'. Cosine itself is a basic trigonometric function that defines the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Therefore, \( \cos^2(x) \) essentially goes a step further by squaring this ratio. This function plays a key role in various trigonometric identities, including the fundamental Pythagorean identity.

In the context of the exercise, \( \cos^2(\beta) \) signifies the square of the cosine of the angle \( \beta \), which is one of the parts of the given identity to be verified. This forms the basis on which we analyze and understand the relations between different trigonometric functions.

Sine Squared

Similar to 'cosine squared', the term 'sine squared’ represented as \( \sin^2(x) \), indicates the square of the sine of an angle 'x'. The sine function measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. Squaring this ratio gives us the \( \sin^2(x) \) function, which is equally important in the realm of trigonometric identities.

By understanding both \( \cos^2(x) \) and \( \sin^2(x) \), one is able to see how these functions interact and complement each other within the framework of trigonometric equations. In the solved problem, we see the transformation of \( \cos^2\left(\frac{\pi}{2} - \beta\right) \) into \( \sin^2(\beta) \), which is a strategic step in verifying the original identity by taking advantage of known relationships between cosine and sine.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry and is crucial for verifying various trigonometric equations. This identity states that for any angle \( \theta \), the sum of the squares of sine and cosine of that angle equals one: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \].

This equation is a trigonometric version of the Pythagorean theorem, which relates the sides of a right-angled triangle. The identity is critical for simplifying trigonometric expressions and solving equations. In our specific exercise, the Pythagorean identity is used in Step 3 to combine \( \cos^2(\beta) \) and the previously obtained \( \sin^2(\beta) \), arriving at the elegantly simplified truth that their sum equals 1, thus verifying the given identity and demonstrating a profound connection among the trigonometric functions.

Complementary Angles

Complementary angles are two angles whose sum amounts to \( \frac{\pi}{2} \) radians (or 90 degrees). This concept is incredibly useful in trigonometry because certain trigonometric functions of complementary angles have special relationships. For instance, the sine of an angle is equal to the cosine of its complement, and vice versa, which is expressed as \( \sin(\alpha) = \cos\left(\frac{\pi}{2} - \alpha\right) \).

In the context of our problem, the variables \( \alpha \) and \( \beta \) are introduced to represent complementary angles, where \( \alpha = \frac{\pi}{2} - \beta \). This relationship is what allows us to replace \( \cos^2\left(\frac{\pi}{2} - \beta\right) \) with \( \sin^2(\beta) \), taking advantage of the intrinsic symmetry and aiding in simplifying the original trigonometric identity. Understanding complementary angles is essential for solving a wide array of trigonometric problems, such as the one we've dissected here.