Problem 24 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Proved: \( \csc \left( \frac{\pi}{2} - u \right) = \sec u \).

Step 1: Recall the Cofunction Identity

The cofunction identity we want to prove is \( \csc \left( \frac{\pi}{2} - u \right) = \sec u \). We will use trigonometric identities and substitution strategies to demonstrate this proof.

Step 2: Express in Terms of Sine

Recall the definition of cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). So, \( \csc \left( \frac{\pi}{2} - u \right) = \frac{1}{\sin \left( \frac{\pi}{2} - u \right)} \).

Step 3: Use the Sine Subtraction Formula

Utilize the sine subtraction formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Here, substitute \( a = \frac{\pi}{2} \) and \( b = u \). This gives us \( \sin \left( \frac{\pi}{2} - u \right) = \sin \frac{\pi}{2} \cos u - \cos \frac{\pi}{2} \sin u \).

Step 4: Simplify the Expression

We know that \( \sin \frac{\pi}{2} = 1 \) and \( \cos \frac{\pi}{2} = 0 \). Substitute these values to get: \( \sin \left( \frac{\pi}{2} - u \right) = 1 \cdot \cos u - 0 \cdot \sin u = \cos u \).

Step 5: Substitute Back to the Identity

Substitute back into the cosecant expression: \( \csc \left( \frac{\pi}{2} - u \right) = \frac{1}{\cos u} \), which is the definition of secant, \( \sec u \). Therefore, \( \csc \left( \frac{\pi}{2} - u \right) = \sec u \).

Key Concepts

Cofunction Identities

Cofunction identities are an essential part of trigonometry that relate the trigonometric functions of complementary angles. Complementary angles add up to \( \frac{\pi}{2} \) radians, or 90 degrees. Such relationships allow one to express a trigonometric function in terms of a different one based on these complementary angles.
For example, a key cofunction identity is \( \sin\left(\frac{\pi}{2} - u\right) = \cos u \). Here, sine of an angle can be expressed not just as sine but as a cosine function of its complement.
This cofunction nature is valuable for simplifying expressions or solving equations involving trigonometric functions.

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

These identities enable switches between functions that simplify calculations, particularly in proofs and integration.

Sine Subtraction Formula

The sine subtraction formula is a pivotal trigonometric identity that helps to find the sine of the difference between two angles. It is written as: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
This formula decomposes the sine of a difference into a product of sines and cosines, which makes it handy when evaluating the sine of complex angles.
For instance, in proving the identity \( \csc\left(\frac{\pi}{2} - u\right) = \sec u \), substituting \( a = \frac{\pi}{2} \) and \( b = u \) shows how straightforward it can simplify an expression.

• For this specific example, \( \sin \frac{\pi}{2} = 1 \) and \( \cos \frac{\pi}{2} = 0 \).
• Thus, \( \sin\left(\frac{\pi}{2} - u\right) = 1 \cdot \cos u - 0 \cdot \sin u = \cos u \).

This clarity transforms the problem into familiar trigonometric functions.

Cosecant and Secant Identities

Understanding the identities of cosecant and secant is crucial in working with inverse trigonometric functions.
The cosecant function is the reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \), while the secant function is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
When working with expressions like \( \csc\left(\frac{\pi}{2} - u\right) \), recognizing that this can be transformed using the sine subtraction formula to \( \cos u \) is pivotal.

• This step then naturally leads to the identification with the secant, due to the definition \( \sec u = \frac{1}{\cos u} \).
• Thus, the initial expression simplifies down to being equal to \( \sec u \) through known identities.

The reciprocal relationships elucidate fundamental properties that help in converting between forms and proving identities.