Problem 23 - Precalculus Mathematics for Calculus

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 identity \( \sec\left(\frac{\pi}{2}-u\right)=\csc u \) is proven by applying the subtraction formula for cosine and simplifying.

Step 1: Recall the Addition Formula for Cosine

The addition formula for cosine is given by \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). In our problem, we want to use the subtraction form: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

Step 2: Apply the Subtraction Formula to Secant

Since \( \sec(x) = \frac{1}{\cos(x)} \), apply the formula to \( \cos\left(\frac{\pi}{2} - u\right) \). Thus, \[ \cos\left(\frac{\pi}{2} - u\right) = \cos\left(\frac{\pi}{2}\right)\cos(u) + \sin\left(\frac{\pi}{2}\right)\sin(u) \].

Step 3: Evaluate Trigonometric Values

We know \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \). Substitute these values into the expression: \[ \cos\left(\frac{\pi}{2} - u\right) = 0 \cdot \cos(u) + 1 \cdot \sin(u) = \sin(u) \].

Step 4: Find the Secant Expression

Now find \( \sec\left(\frac{\pi}{2} - u\right) \) by taking the reciprocal: \[ \sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\cos\left(\frac{\pi}{2} - u\right)} = \frac{1}{\sin(u)} = \csc(u) \].

Step 5: Conclude the Proof

Since we have shown that \( \sec\left(\frac{\pi}{2} - u\right) = \csc(u) \), the cofunction identity is proven using the addition and subtraction formulas.

Key Concepts

Cofunction Identity

The concept of cofunction identity in trigonometry is fascinating as it relates complementary angles. Cofunction identities demonstrate the relationship between different trigonometric functions, tuned to angles that add up to a right angle, \( \frac{\pi}{2} \ \text{or} \ 90^{\circ} \). This relationship states that the trigonometric function of an angle is equal to the cofunction of its complement.
In simpler terms:

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

These identities are pivotal in simplifying expressions and solving equations in trigonometry. They help connect trigonometric expressions of different angles, providing insights into the nature of these functions.

Addition Formula for Cosine

The addition formula for cosine is crucial when dealing with trigonometric problems involving sums of angles. It provides a way to express the cosine of the sum of two angles, and is given by:

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

This formula allows us to find the cosine of an angle that is composed of two other angles. It's applicable in various contexts, such as simplifying expressions or solving trigonometric equations.
Understanding this formula is essential for more complex problems, as it sets the foundation for comprehending the relationships between trigonometric functions. It's often used alongside subtraction formulas, especially when proving identities or transforming expressions.

Subtraction Formula

The subtraction formula for cosine is another significant tool in trigonometry. It helps express the cosine of the difference between two angles. The formula is:

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

This formula is particularly useful when you need to relate the cosine of one angle minus another. The subtraction formula can convert a subtraction problem into simpler components by breaking it down into basic trigonometric functions.
For example, in proving the cofunction identity \( \sec\left(\frac{\pi}{2} - u\right) = \csc(u) \), we employ this formula to handle the angle subtraction, changing it into individual sine and cosine terms that are easier to manage.

Secant

Secant is a trigonometric function that complements the cosine function. Given as the reciprocal of cosine, secant is used in various trigonometric equations and identities. Its definition is:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Secant is particularly important in identities where cosine reaches zero, as it helps avoid undefined expressions by considering reciprocal relationships.
In the context of the given exercise, the secant function plays a central role as we explore its cofunction identity with cosecant. By rearranging the subtraction formula, we can express secant in terms relevant to the exercise, emphasizing the relation with an angle's complement.

Cosecant

Cosecant is the reciprocal of sine, crucial in various trigonometric contexts, such as in identities and equations. Defined as:

• \( \csc(x) = \frac{1}{\sin(x)} \)

Cosecant is used when dealing with angles or expressions where sine is minimal or reaches zero, ensuring a non-zero evaluation by flipping the function.
In the cofunction identity \( \sec\left(\frac{\pi}{2} - u\right) = \csc(u) \), cosecant provides a harmonic view of the same identity, revealing how complementary angles relate their reciprocal ratios in trigonometric terms.