Problem 19 - Algebra and Trigonometry

Question

\(19-22\) . Prove the cofunction identity using the addition and subtraction formulas. $$ \tan \left(\frac{\pi}{2}-u\right)=\cot u \quad \text { 20. } \cot \left(\frac{\pi}{2}-u\right)=\tan u $$

Step-by-Step Solution

Answer

\( \tan \left( \frac{\pi}{2} - u \right) = \cot u \) and \( \cot \left( \frac{\pi}{2} - u \right) = \tan u \).

Step 1: State the Cofunction Identity

The cofunction identity you need to prove states: \( \tan \left( \frac{\pi}{2} - u \right) = \cot u \). This means the tangent of the complement of angle \( u \) is equal to the cotangent of \( u \).

Step 2: Use the Tangent Subtraction Formula

Recall the tangent subtraction formula: \( \tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). Apply this to \( \tan \left( \frac{\pi}{2} - u \right) \), where \( a = \frac{\pi}{2} \) and \( b = u \).

Step 3: Compute Tangent at \(\frac{\pi}{2}\)

The tangent of \( \frac{\pi}{2} \) is undefined, which poses a problem. To address this, recall that the tangent of a complementary angle can be rewritten using cotangent: \(\tan \left( \frac{\pi}{2} - u \right) = \frac{1}{\tan u} = \cot u\).

Step 4: Conclude Proof for \( \tan \left( \frac{\pi}{2} - u \right) \)

This shows that according to trigonometric identities and the concept of complementary angles, \( \tan \left( \frac{\pi}{2} - u \right) \) simplifies to \( \cot u \). Hence, the identity \( \tan \left( \frac{\pi}{2} - u \right) = \cot u \) is proven.

Step 5: State the Second Cofunction Identity

The second cofunction identity stated is: \( \cot \left( \frac{\pi}{2} - u \right) = \tan u \). This implies the cotangent of the complement of angle \( u \) equals the tangent of angle \( u \).

Step 6: Use the Cotangent Identity Relation

Recall the cotangent identity \( \cot x = \frac{1}{\tan x} \). Therefore, \( \cot \left( \frac{\pi}{2} - u \right) = \frac{1}{\tan \left( \frac{\pi}{2} - u \right)} \). Using the previously proven identity \( \tan \left( \frac{\pi}{2} - u \right) = \cot u \).

Step 7: Simplify for \( \cot \left( \frac{\pi}{2} - u \right) \)

Substitute \( \tan \left( \frac{\pi}{2} - u \right) \) with \( \cot u \) into the formula: \[ \cot \left( \frac{\pi}{2} - u \right) = \frac{1}{\cot u} = \tan u \].

Step 8: Conclude Proof for \( \cot \left( \frac{\pi}{2} - u \right) \)

Thus, we have proven that \( \cot \left( \frac{\pi}{2} - u \right) = \tan u \), affirming the second identity.

Key Concepts

Tangent Subtraction Formula

The tangent subtraction formula is a fundamental tool in trigonometry. It helps in simplifying expressions involving the tangent of the difference between two angles. The formula is given by: \[ \tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \] To use this formula, you need the tangent of both individual angles. For example, if you need to find \(\tan(\frac{\pi}{2} - u)\), you can set \(a = \frac{\pi}{2}\) and \(b = u\) in the formula. However, since \(\tan(\frac{\pi}{2})\) is undefined, we use a workaround. This is where trigonometric identities and the notion of complementary angles come into play. The tangent of a complementary angle can be expressed in terms of cotangent, simplifying the expression to: \[ \tan(\frac{\pi}{2} - u) = \frac{1}{\tan u} = \cot u \] This relation is key to proving cofunction identities.

Cotangent Identity

The cotangent identity is closely related to the tangent identity and is important in understanding trigonometric functions' interrelations. The basic cotangent identity states: \[ \cot x = \frac{1}{\tan x} \] This means that cotangent is the reciprocal of tangent. When dealing with problems asking you to express one function in terms of the other, this identity is incredibly useful. For instance, to prove cofunction identities, knowing that \(\cot\left( \frac{\pi}{2} - u \right) = \tan u\) hinges on understanding these reciprocal relationships. By substituting \(\tan(\frac{\pi}{2} - u)\) with \(\cot u\), the identity transforms into: \[ \cot\left( \frac{\pi}{2} - u \right) = \frac{1}{\cot u} = \tan u \] This allows us to conclude the cotangent and tangent are cofunctions of complementary angles.

Trigonometric Identities

Trigonometric identities are like bridges among various trigonometric functions. They simplify complex expressions and make solving trigonometric equations much easier. One class of these identities encompasses the cofunction identities, which express a function of an angle in terms of a function of its complement. Examples include:

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

The trigonometric identities of tangent and cotangent are especially useful while dealing with angles formed within right triangles and can be handy for proving mathematical expressions equivalent. Within this context, cofunction identities are special because they relate functions of complementary angles. For instance, proving \( \tan \left( \frac{\pi}{2} - u \right) = \cot u \) utilizes these identities to show that tangent and cotangent are cofunctions of each other. Mastering these identities helps streamline solving a wide range of trigonometric problems.