Problem 21 - Algebra and Trigonometry

Question

\(19-22\) . 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 identity \( \sec\left(\frac{\pi}{2} - u\right) = \csc u \) is proven using subtraction formulas and reciprocal identities.

Step 1: Identify the Cofunction Identity

The cofunction identity we need to prove is \( \sec\left(\frac{\pi}{2} - u\right) = \csc u \). This involves using known trigonometric identities with addition and subtraction formulas.

Step 2: Express Secant in Terms of Cosine

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

Step 3: Use the Cosine Subtraction Formula

The formula for cosine of a difference is given by \( \cos(a-b) = \cos a \cos b + \sin a \sin b \). For \( \cos\left(\frac{\pi}{2} - u\right) \), let \( a = \frac{\pi}{2} \) and \( b = u \).

Step 4: Substitute Values into the Formula

Substitute \( a = \frac{\pi}{2} \) and \( b = u \) into the formula: \( \cos\left(\frac{\pi}{2} - u\right) = \cos \frac{\pi}{2} \cos u + \sin \frac{\pi}{2} \sin u \).

Step 5: Simplify the Expression

We know \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Therefore, \( \cos\left(\frac{\pi}{2} - u\right) = 0 \cdot \cos u + 1 \cdot \sin u = \sin u \).

Step 6: Calculate Secant of the Angle

Now we substitute back into the expression for secant: \( \sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\sin u} \).

Step 7: Recognize the Reciprocal Identities

Recognize that \( \frac{1}{\sin u} = \csc u \). Therefore, \( \sec\left(\frac{\pi}{2} - u\right) = \csc u \), which proves the identity.

Key Concepts

Addition and Subtraction Formulas

The addition and subtraction formulas are key tools for transforming trigonometric expressions. They provide a way to break down expressions involving angles, which makes proving many identities easier. Specifically, the cosine subtraction formula states:

• \( \cos(a-b) = \cos a \cos b + \sin a \sin b \)

This allows us to express the cosine of an angle difference in terms of the basic sine and cosine functions. In the context of our exercise, this formula transforms \( \cos\left(\frac{\pi}{2} - u\right) \). Here, substituting \( a = \frac{\pi}{2} \) and \( b = u \), simplifies \( \cos\left(\frac{\pi}{2} - u\right) \) to \( \sin u \).
Why do we use this formula? Because it reveals a connection between secant and cosecant through the underlying sine function. Understanding addition and subtraction formulas is critical for working with cofunction identities as it helps in manipulating trigonometric expressions to arrive at the desired identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the involved variable. They include fundamental relationships such as those between sine, cosine, tangent, and their respective cofunctions. One of the main goals of using trigonometric identities is to simplify complex expressions and enable the proof of more specific identities.
In our exercise, we utilize the identity \( \sec x = \frac{1}{\cos x} \), which connects secant to cosine. This allows us to transform \( \sec\left(\frac{\pi}{2} - u\right) \) into \( \frac{1}{\sin u} \) by leveraging the earlier result from the cosine subtraction formula.
Understanding such identities expands the toolkit for solving trigonometric problems. They are essential in calculus, engineering, and physics for analyzing periodic phenomena, among many other applications.

Reciprocal Identities

Reciprocal identities are a subset of trigonometric identities where functions are expressed in terms of their reciprocals. These identities are useful for converting functions and simplifying calculations. The main reciprocal identities include:

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

In proving the cofunction identity \( \sec\left(\frac{\pi}{2} - u\right) = \csc u \), recognizing that \( \frac{1}{\sin u} = \csc u \) is crucial. This reciprocal relationship shows that secant of a cofunction angle corresponds directly to cosecant.
This insight is part of why understanding reciprocal identities is so important: they provide a bridge to connect and transition between different trigonometric functions, simplifying complex proofs and expressions.