Problem 24 - Algebra and Trigonometry

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

The identity is proven by showing \( \csc \left(\frac{\pi}{2} - u\right) = \frac{1}{\cos u} = \sec u \).

Step 1: Understand the Cofunction Identity

The cofunction identity you need to prove is \( \csc \left(\frac{\pi}{2}-u\right)=\sec u \). This means that the cosecant of \( \frac{\pi}{2}-u \) equals the secant of \( u \).

Step 2: Recall the Cosecant and Secant Definitions

The cosecant function is defined as \( \csc x = \frac{1}{\sin x} \). The secant function is defined as \( \sec x = \frac{1}{\cos x} \). We aim to show that \( \csc \left(\frac{\pi}{2}-u\right) \) and \( \sec u \) are equal.

Step 3: Use the Sine and Cosine Cofunction Identities

The cofunction identities state that \( \sin \left(\frac{\pi}{2} - u\right) = \cos u \) and \( \cos \left(\frac{\pi}{2} - u\right) = \sin u \). Use these to rewrite the expression.

Step 4: Substitute and Simplify

Substitute the identity \( \sin \left(\frac{\pi}{2} - u\right) = \cos u \) into the cosecant definition: \[ \csc \left(\frac{\pi}{2} - u\right) = \frac{1}{\sin \left(\frac{\pi}{2} - u\right)} = \frac{1}{\cos u} \].

Step 5: Recognize the Equivalent Expression

Notice that \( \frac{1}{\cos u} = \sec u \). This shows that \( \csc \left(\frac{\pi}{2} - u\right) \) is indeed equal to \( \sec u \).

Step 6: Conclude the Proof

Through substitution and simplification using known trigonometric identities, we have proven that \( \csc \left(\frac{\pi}{2} - u\right) = \sec u \).

Key Concepts

Cofunction Identity

The cofunction identity is a fundamental concept in trigonometry, particularly when dealing with angle transformations. Cofunction identities reveal the relationship between specific trigonometric functions of complementary angles. For instance, the angles \( rac{\pi}{2} - u \) and \( u \) are complements, meaning they add up to \( \frac{\pi}{2} \) (or 90 degrees).
Applying this to trigonometric functions, the cofunction identity for cosecant and secant states that \( \csc \left(\frac{\pi}{2} - u\right) = \sec u \). This indicates that the cosecant of one angle equals the secant of its complementary angle.
Understanding this relationship helps simplify expressions and solve trigonometric equations more efficiently, especially in trigonometry problems involving complementary angles. Using cofunction identities allows for transformation between functions, targeting simplification and solving for unknowns with ease.
When these identities are combined with proper algebraic handling, they reveal deep insights into symmetrical properties and help streamline complex calculations.

Addition and Subtraction Formulas

Addition and subtraction formulas are instrumental in trigonometry for expressing and solving expressions involving sums or differences of angles. These formulas provide a way to decompose trigonometric functions of multiple angles into products of trigonometric functions of single angles.
The key formulas include:

• \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)
• \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)

Applying these, the sine and cosine cofunction identities can be derived. Using \( \sin \left( \frac{\pi}{2} - u \right) = \cos u \) from the subtraction formula, leads directly into forming other identities like the one involving cosecant and secant.
These transformations greatly improve the ability to manipulate trigonometric expressions, allowing for the proof of relationships like \( \csc \left( \frac{\pi}{2} - u \right) = \sec u \). It's all about breaking down complex angles into something simpler, which is foundational for solving trigonometric equations in calculus and beyond.

Cosecant and Secant Functions

Cosecant and secant are important reciprocal trigonometric functions related closely to sine and cosine, respectively. Understanding their definitions is essential for grasping more complex trigonometric identities.
The cosecant function is defined as \( \csc x = \frac{1}{\sin x} \), highlighting its reciprocal nature to the sine function. Similarly, the secant function is defined as \( \sec x = \frac{1}{\cos x} \), showing it as the reciprocal of cosine.
These functions are particularly useful when dealing with ratios and angles as opposed to focusing purely on side lengths, which is often the case with sine and cosine. By using the relationship of the cofunction identity between these, such as \( \csc \left( \frac{\pi}{2} - u \right) = \sec u \), it becomes evident how one function can describe another based on the angle's complement.
Such identities and transformations are not only helpful for simplifying equations but also serve to tie together various areas of trigonometric study, making them accessible and solve real-world problems efficiently.