Problem 23 - Algebra and Trigonometry

Question

Prove the cofunction identity using the Addition and Subtraction Formulas. $$ \sec \left(\frac{\pi}{2}-u\right)=\csc u $$

Step-by-Step Solution

Answer

The cofunction identity \( \sec(\frac{\pi}{2}-u) = \csc(u) \) is verified using trigonometric identities.

Step 1: Recall the Cofunction Identity for Sine and Cosine

The cofunction identity for secant and cosecant is based on the complementary angle identity of sine and cosine: \( \sin \left(\frac{\pi}{2} - u\right) = \cos(u) \) and \( \cos\left(\frac{\pi}{2} - u\right) = \sin(u) \).

Step 2: Write the Secant Function in Terms of Cosine

Remember that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, \( \sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\cos\left(\frac{\pi}{2} - u\right)} \).

Step 3: Apply the Cosine Cofunction Identity

Using the cofunction identity for cosine, \( \cos\left(\frac{\pi}{2} - u\right) = \sin(u) \). Substitute this into the secant expression: \( \sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\sin(u)} \).

Step 4: Recognize the Reciprocal Identity for Cosecant

Recall that \( \csc(u) = \frac{1}{\sin(u)} \). Thus, the expression \( \frac{1}{\sin(u)} \) is equivalent to \( \csc(u) \).

Step 5: Conclude the Identity

Combine the results from the previous steps to verify the identity: \( \sec\left(\frac{\pi}{2} - u\right) = \csc(u) \).

Key Concepts

Cofunction Identities

Cofunction identities in trigonometry reveal the relationship between a function and its cofunction. Two angles are known as cofunctions if their sum equals \( \frac{\pi}{2} \) or 90 degrees. For sine and cosine, this translates into:

• \( \sin\left(\frac{\pi}{2} - u\right) = \cos(u) \)
• \( \cos\left(\frac{\pi}{2} - u\right) = \sin(u) \)

These relationships help in proving identities like the one in the exercise. Here, secant and cosecant utilize these identities to establish their connection. By understanding these identities, it becomes easier to manipulate and transform trigonometric expressions to reveal deeper truths about angles and their relationships.

Secant Function

The secant function is one of the reciprocal trigonometric functions. It is the reciprocal of the cosine function, represented as: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). In terms of angle transformations, secant can be adjusted using cofunction identities. This function is very helpful in analyzing triangles and waves, especially where angles are close to 90 degrees or \( \frac{\pi}{2} \). In the context of the current problem, we see its evaluation at \( \frac{\pi}{2} - u \), showing how secant utilizes cosine's complementary values. Secant's dependency on the cosine helps in proving various trigonometric identities by revealing the inverse relationship.

Cosecant Function

Similar to secant, the cosecant function is also a reciprocal trigonometric function. It is specifically the reciprocal of the sine function: \( \csc(u) = \frac{1}{\sin(u)} \). Cosecant is often used in situations involving right triangles or periodic functions, where sine is zero or undefined. In the exercise, understanding that \( \sec\left(\frac{\pi}{2} - u\right) = \csc(u) \) shows how reciprocal and complementary relationships among trig functions can simplify expressions. This proves particularly useful in calculus and geometry, where angle manipulation is key.

Reciprocal Identities

Reciprocal identities are essential in trigonometry. These identities express trigonometric functions as the reverse of another fundamental function:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

In the given exercise, these identities allow us to switch between different trigonometric expressions easily. Understanding and applying reciprocal identities assist students in solving equations and proving identities. They reveal how functions are related inversely; for example, recognizing that turning secant into a cosecant already uses the reciprocal nature of these trigonometric functions.