Prove the cofunction identity using the Addition and Subtraction Formulas.

sec

𝜋

2

− *u* = csc(*u*)

Use a Reciprocal Identity, then apply a Subtraction Formula to simplify.

sec

𝜋

2

− *u*

=

1

cos

=

1

+ sin

𝜋

2

sin(*u*)

=

1

· cos(*u*) +

· sin(*u*)

=

1

=

To prove the cofunction identity 

\[
\sec\left(\frac{\pi}{2} - u\right) = \csc(u)
\]

using the reciprocal and subtraction identities, we will follow these steps:

### Step 1: Rewrite \(\sec\left(\frac{\pi}{2} - u\right)\) using the Reciprocal Identity
The secant function is defined as the reciprocal of the cosine function, so we can write:

\[
\sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\cos\left(\frac{\pi}{2} - u\right)}
\]

### Step 2: Apply the Cofunction Identity for Cosine
Using the subtraction formula, we know that:

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

This follows from the cofunction identity, which states that \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\) for any angle \(x\).

Thus,

\[
\sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\sin(u)}
\]

### Step 3: Recognize the Result as the Definition of Cosecant
By definition, \(\csc(u) = \frac{1}{\sin(u)}\). Therefore,

\[
\sec\left(\frac{\pi}{2} - u\right) = \csc(u)
\]

### Conclusion
We have shown that:

\[
\sec\left(\frac{\pi}{2} - u\right) = \csc(u)
\]

This completes the proof using the reciprocal identity and the subtraction formula.

---

Would you like a breakdown of each step or have any questions? Here are a few related questions for deeper understanding:

1. What are other examples of cofunction identities in trigonometry?
2. How do the reciprocal identities relate to cofunction identities?
3. Can you prove a similar identity for \(\sin\left(\frac{\pi}{2} - u\right) = \cos(u)\)?
4. How are addition and subtraction formulas generally applied in trigonometric proofs?
5. What role does the unit circle play in understanding these identities?

**Tip:** Remember that cofunction identities relate trigonometric functions of complementary angles.

Prove the cofunction identity using the Addition and Subtraction Formulas.

sec(π/2 - u) = csc(u)

Use a Reciprocal Identity, then apply a Subtraction Formula to simplify.

Learn how to prove the cofunction identity sec(π/2 - u) = csc(u) through a detailed step-by-step solution using reciprocal and subtraction identities. This solution explores trigonometric concepts, including cofunction and reciprocal identities, ideal for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation