Problem 22 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\cot\left(\frac{\pi}{2} - u\right) = \tan u\) is correct using angle subtraction and identities.

Step 1: Recall Cofunction Identity

The cofunction identity we need to prove states that \(\cot \left(\frac{\pi}{2}-u\right)=\tan u\). We will use the addition and subtraction formulas involving trigonometric functions to prove this identity.

Step 2: Use Cotangent Identity

Recall that the cotangent of an angle \(x\) is the reciprocal of the tangent: \(\cot x = \frac{1}{\tan x}\). This will help to transition between cotangent and tangent.

Step 3: Apply Cotangent Angle Difference Formula

The angle difference identity for cotangent is:\[\cot(A - B) = \frac{\cot A \cot B + 1}{\cot B - \cot A}\]Substitute \(A = \frac{\pi}{2}\) and \(B = u\).

Step 4: Evaluate Angles for Cotangent

We know that \(\cot \frac{\pi}{2} = 0\) (since \(\tan \frac{\pi}{2}\) is undefined and \(\cot \frac{\pi}{2}\) is zero). Therefore, substitute into the identity:\[\cot\left(\frac{\pi}{2} - u\right) = \frac{0 \cdot \cot u + 1}{\cot u - 0} = \frac{1}{\cot u}\]

Step 5: Use Reciprocal Identity

Recall that \(\tan u = \frac{1}{\cot u}\). Substitute this into our equation to get:\[\cot\left(\frac{\pi}{2} - u\right) = \tan u\]

Step 6: Conclusion

By using the angle subtraction formula for cotangent and the reciprocal identities, we have shown that \(\cot\left(\frac{\pi}{2} - u\right) = \tan u\), thus proving the given cofunction identity.

Key Concepts

Cofunction Identities

Cofunction identities are special relationships in trigonometry that connect pairs of trigonometric functions whose inputs are complementary angles. Complementary angles add up to \( \frac{\pi}{2} \) radians (or 90 degrees). For example, the sine and cosine functions are cofunctions, which means that \( \sin(\frac{\pi}{2} - u) = \cos(u) \). Similarly, the cotangent and tangent functions are cofunctions; so, \( \cot(\frac{\pi}{2} - u) = \tan u \) belongs to these cofunction identities.

• Definition: Two angles are complementary if their sum is \( \frac{\pi}{2} \).
• Common Cofunction Relationships: For sine and cosine: \( \sin(\frac{\pi}{2} - u) = \cos u \), and for tangent and cotangent: \( \tan(\frac{\pi}{2} - u) = \cot u \).
• Application: Cofunction identities often simplify calculations by transforming one trigonometric function into another.

Understanding these identities allows us to relate and convert between different trigonometric functions effectively, making calculations easier and more intuitive.

Addition and Subtraction Formulas

The addition and subtraction formulas in trigonometry are tools for finding trigonometric function results based on the sums or differences of angles. These formulas help rewrite expressions where two angles are involved. They work for sine, cosine, tangent, and cotangent functions. A typical form is \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \).

Using the addition and subtraction formulas, we can simplify trigonometric identities, which involve complex angle manipulations. For example, the formula for tangent difference is:

• Tangent of Difference: \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)
• Cotangent of Difference: \( \cot(A - B) = \frac{\cot A \cot B + 1}{\cot B - \cot A} \)

In the given problem, this formula helps us in transforming \( \cot(\frac{\pi}{2} - u) \) into a form that reveals it as \( \tan u \). This demonstrates how these formulas bridge gaps between angles and trigonometric functions, profoundly aiding in identity proofs.

Cotangent and Tangent Relationships

Cotangent and tangent are closely related trigonometric functions, being reciprocals of each other. The basic relationship between these functions is given by \( \cot x = \frac{1}{\tan x} \) and \( \tan x = \frac{1}{\cot x} \). This reciprocal relationship is critical in simplifying trigonometric expressions involving these functions.

• Reciprocity: Remember, \( \cot x \) is the reciprocal of \( \tan x \), which means if you know one, you can find the other.
• Application in Identities: The reciprocal nature helps in proving identities like \( \cot(\frac{\pi}{2} - u) = \tan u \), where using \( \tan u = \frac{1}{\cot u} \) can simplify and prove expressions.

By leveraging the reciprocal relationship, you can handle complex trigonometric problems with greater ease, especially when converting or proving identities.