Problem 46 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \sec^2 \left(\dfrac{\pi}{2} - x \right) - 1 = \cot^2 x \)

Step-by-Step Solution

Answer

Both sides have been proven to be equivalent, thus verifying the identity \( \sec^2 \left(\frac{\pi}{2} - x \right) - 1 = \cot^2 x \)

Step 1: Express the left-hand side in terms of sine or cosine

This is done by using the definition of secant, which is \( \sec x = \frac{1}{\cos x} \). Thus, \( \sec^2 \left(\frac{\pi}{2} - x \right) = \frac{1}{\cos^2 \left(\frac{\pi}{2} - x \right)} \). Using sine and cosine transformations, we know \( \cos \left(\frac{\pi}{2} - x \right) = \sin x \). So, this further simplifies the left-hand side to \( \frac{1}{\sin^2 x} \).

Step 2: Simplify the left-hand side further

To make the equation easier to work with, subtract 1: \( \frac{1}{\sin^2 x} - 1 \). This can be rewritten as \( \frac{1 - \sin^2 x}{\sin^2 x} \).

Step 3: Convert into cotangent formula

Now use the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \) to simplify the equation to \( \frac{\cos^2 x}{\sin^2 x} \). Using the definition of cotangent, \( \cot x = \frac{\cos x}{\sin x} \), we find \( \frac{\cos^2 x}{\sin^2 x} = \cot^2 x \). So we have successfully shown that the left-hand side equals the right-hand side.

Key Concepts

Secant Function

In trigonometry, the secant function is an important identity used to express angles in terms of a right triangle. The secant of an angle, denoted as \( \sec x \), is the reciprocal of the cosine function. This means \( \sec x = \frac{1}{\cos x} \). Understanding this relationship is key when simplifying expressions involving trigonometric identities.In the context of trigonometric transformations, one often needs to express secant in terms of sine or cosine to facilitate simplifications. For example, in our original exercise, \( \sec^2 \left(\frac{\pi}{2} - x \right) \) was rewritten using the cosine function. This substitution often helps to transform complex trigonometric expressions into simpler, more manageable forms.

Cotangent Function

The cotangent function, \( \cot x \), is another trigonometric identity that's frequently used in trigonometric solutions and transformations.The cotangent is the reciprocal of the tangent function, expressed as \( \cot x = \frac{1}{ an x} = \frac{ ext{adjacent}}{ ext{opposite}} = \frac{ ext{cosine}}{ ext{sine}} \). This identity is particularly useful when the goal is to simplify expressions involving ratios of sine and cosine. As demonstrated in the given exercise, an expression involving the cotangent function was derived by reorganizing a more complex secant-based expression using sine and cosine functions.When simplifying or verifying identities, converting expressions to involve cotangent can offer clearer insights into their equivalences.

Pythagorean Identity

One of the most fundamental identities in trigonometry is the Pythagorean identity. It states that for any angle \( x \), the following is always true: \( \sin^2 x + \cos^2 x = 1 \).This identity stems from the Pythagorean theorem and is a cornerstone for proving many other trigonometric identities. By rearranging this identity, you can express \( \cos^2 x = 1 - \sin^2 x \), which was utilized in the exercise to transform the expression \( \frac{1 - \sin^2 x}{\sin^2 x} \) into \( \frac{\cos^2 x}{\sin^2 x} \).Understanding and applying the Pythagorean identity allows for the simplification of expressions and the verification of trigonometric identities by transforming trigonometric functions into their equivalent forms.

Sine and Cosine Transformations

Sine and cosine transformations are techniques used extensively in trigonometry to convert expressions into a form that's easier to work with.A common transformation involves the complementary angle identity: \( \cos \left(\frac{\pi}{2} - x \right) = \sin x \) and \( \sin \left(\frac{\pi}{2} - x \right) = \cos x \). These transformations are pivotal when you need to simplify or unify expressions, especially when dealing with angles expressed in different forms.In the provided exercise, the cosine of complementary angle was transformed into a sine function, specifically \( \cos \left(\frac{\pi}{2} - x \right) = \sin x \). Understanding and applying these transformations can significantly aid in resolving complex trigonometric equations and identities.