Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Derive the formula \(\cot (u+v)=\frac{\cot u \cot v-1}{\cot u+\cot v}\).

Step-by-Step Solution

Answer

The formula is derived by applying angle sum identities and trigonometric identities to simplify and express \(\cot(u+v)\) in terms of \(\cot u\) and \(\cot v\).

Step 1: Understand the Problem

The given problem asks us to derive the formula for the cotangent of the sum of two angles, \(u\) and \(v\). We need to express \(\cot (u+v)\) in terms of \(\cot u\) and \(\cot v\).

Step 2: Recall Trigonometric Identities

To begin, recall the identity for cotangent: \(\cot x = \frac{\cos x}{\sin x}\). Also, use the angle sum identities: \(\sin(u+v) = \sin u \cos v + \cos u \sin v\) and \(\cos(u+v) = \cos u \cos v - \sin u \sin v\).

Step 3: Express \(\cot(u+v)\) Using Sine and Cosine

Write \(\cot(u+v)\) as \(\frac{\cos(u+v)}{\sin(u+v)}\). Substitute the angle sum identities to get \[ \cot(u+v) = \frac{\cos u \cos v - \sin u \sin v}{\sin u \cos v + \cos u \sin v}. \]

Step 4: Substitute \(\cot u\) and \(\cot v\)

Since \(\cot u = \frac{\cos u}{\sin u}\) and \(\cot v = \frac{\cos v}{\sin v}\), express \(\cos u\) and \(\cos v\) as \(\cos u = \cot u \sin u\) and \(\cos v = \cot v \sin v\). Substitute these into the equation for \(\cot(u+v)\):\[ \cot(u+v) = \frac{(\cot u \sin u)(\cot v \sin v) - \sin u \sin v}{\sin u (\cot v \sin v) + \sin v (\cot u \sin u)}. \]

Step 5: Simplify the Expression

Simplify the expression by factoring out \(\sin u \sin v\) from both numerator and denominator. This gives:\[ \cot(u+v) = \frac{(\cot u \cot v - 1)\sin u \sin v}{(\cot v \sin u + \cot u \sin v) \sin u \sin v}. \]

Step 6: Final Simplification

Cancel \(\sin u \sin v\) from the numerator and the denominator to arrive at the desired formula:\[ \cot(u+v) = \frac{\cot u \cot v - 1}{\cot u + \cot v}. \]

Step 7: Verify and Conclude

Verify each step and conclude that the derived formula is consistent with the properties of trigonometric functions. The steps have yielded the intended formula of \(\cot(u+v) = \frac{\cot u \cot v - 1}{\cot u + \cot v}.\)

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics to simplify expressions and solve complex trigonometric equations. These identities express relationships between different trigonometric functions, like sine, cosine, and tangent. Some of the most commonly used identities include the Pythagorean identities, such as \( \sin^2(x) + \cos^2(x) = 1 \), and reciprocal identities, such as \( \csc(x) = \frac{1}{\sin(x)} \).

These identities allow us to transform complex trigonometric expressions into simpler forms. For example, utilizing the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) can make integration tasks more manageable. Trigonometric identities are fundamental in calculus and other advanced areas of mathematics. They are crucial for solving problems involving periodic functions and wave phenomena, bridging geometry and algebra.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the tangent function: \( \cot(x) = \frac{1}{\tan(x)} \). Since the tangent function itself is the ratio of sine to cosine, we have \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). This shows that the function is undefined wherever \( \sin(x) = 0 \), meaning every \( x \) where \( x = n \pi \) for integer \( n \).

Cotangent possesses its unique properties and behaviors. It is periodic with a period of \( \pi \) and has vertical asymptotes wherever sine is zero. Understanding cotangent is pivotal in trigonometry as it appears in various applications, such as in the analysis of slopes and angles in coordinate geometry. When plotting, one will notice the function falls rapidly within its primary interval, differentiating it significantly from its counterpart, the tangent function.

Angle Sum Identities

Angle sum identities allow the evaluation of trigonometric functions for the sums (and differences) of two angles. These identities are particularly useful in deriving other trigonometric formulas and solving trigonometric equations. The main angle sum identities include \( \sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v) \) and \( \cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v) \).

By knowing these identities, we can find the sine, cosine, or tangent of non-standard angles by breaking them down into known angles. For instance, calculating \( \sin(75^\circ) \) using \( \sin(45^\circ + 30^\circ) \) would necessitate these identities.

• This technique helps reduce complexity in solving trigonometric problems by relying on the values of more common angles like \( 30^\circ, 45^\circ, \) and \( 60^\circ \).
• These identities also serve as building blocks for deriving further relations, such as the double angle or half-angle identities, expanding their usability across different branches of mathematics.

Understanding and using angle sum identities is crucial in areas involving rotation and oscillation, such as physics and engineering dynamics.