Problem 24 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\cot 2 u=\frac{\cot ^{2} u-1}{2 \cot u}\)

Step-by-Step Solution

Answer

The identity \( \cot 2u = \frac{\cot^2 u - 1}{2 \cot u} \) is verified.

Step 1: Recall Trigonometric Identities

Start by recalling the trigonometric identities for cotangent and double angle formulae. We know that \( \cot 2u = \frac{\cos 2u}{\sin 2u} \). We also know the double angle formulas: \( \cos 2u = \cos^2 u - \sin^2 u \) and \( \sin 2u = 2 \sin u \cos u \).

Step 2: Express \( \cot 2u \) Using Known Identities

Substitute the double angle formulas into the expression for \( \cot 2u \): \[ \cot 2u = \frac{\cos^2 u - \sin^2 u}{2 \sin u \cos u} \].

Step 3: Simplify the Cotangent Expression

Convert \( \cos^2 u - \sin^2 u \) using the identity \( \cos^2 u - \sin^2 u = (\cos u + \sin u)(\cos u - \sin u) \). So: \[ \cot 2u = \frac{(\cos u + \sin u)(\cos u - \sin u)}{2 \sin u \cos u} \].

Step 4: Simplify to Match Target Expression

Rewrite the right side: \( \cot^2 u - 1 = (\frac{\cos u}{\sin u})^2 - 1 = \frac{\cos^2 u - \sin^2 u}{\sin^2 u} \). Verify that this fraction can be separated and that the numerator can be divided by the denominator, matching with \( 2\cot u \).

Step 5: Verification of the Simplification

Combine all components to show: \[ \frac{\cot^2 u - 1}{2\cot u} = \frac{\frac{\cos^2 u - \sin^2 u}{\sin^2 u}}{2\frac{\cos u}{\sin u}} = \frac{(\cos^2 u - \sin^2 u)}{2 \sin u \cos u} \]. Thus, both expressions simplify identically.

Key Concepts

Double Angle Formulas

Double angle formulas are an essential part of trigonometry that allows us to express trigonometric functions of double angles, like \(2u\), in terms of single angles, this helps simplify complex trigonometric equations.
\[\]Some popular double angle formulas include:

• The cosine double angle formula: \( \cos 2u = \cos^2 u - \sin^2 u \). This can also be rewritten as \( \cos 2u = 2\cos^2 u - 1 \) or \( \cos 2u = 1 - 2\sin^2 u \).


• The sine double angle formula: \( \sin 2u = 2 \sin u \cos u \).


• The tangent double angle formula: \( \tan 2u = \frac{2 \tan u}{1 - \tan^2 u} \).

In the problem, we use \( \cos 2u \) and \( \sin 2u \) double angle formulas to move from \(\cot 2u\) expressed in terms of sine and cosine of double angles, preparing it to be simplified into the target expression. Understanding double angle formulas makes these conversions and simplifications much more manageable.

Cotangent Function

The cotangent function, denoted as \(\cot\), is one of the basic trigonometric functions. It is related to the tangent function, and is indeed the reciprocal of tangent.
\[\]In terms of sine and cosine, it is expressed as:

• \( \cot u = \frac{\cos u}{\sin u} \).

The functionality of \(\cot\) directly relates to its purpose of describing the connection between the adjacent side and the opposite side of a right-angled triangle.
\[\]Within the exercise, \( \cot^2 u - 1 \) appears when verifying identities, showcasing how \(\cot\) interacts with other functions when manipulating expressions. Recognizing how to rewrite expressions using \(\cot\) is crucial, particularly in verifying or simplifying trigonometric identities.
\[\]The cotangent double angle, \( \cot 2u \), provided here in the identity \( \cot 2u = \frac{\cot ^{2} u-1}{2 \cot u} \), further demonstrates the role of \(\cot\) in complex trigonometric simplification and verification.

Trigonometric Simplification

Trigonometric simplification is an invaluable skill in mathematics, paving the way to solve more complex problems by breaking them into manageable pieces.
\[\]When given a trigonometric identity, such as \( \cot 2u = \frac{\cot ^{2} u-1}{2 \cot u} \), simplification involves several steps:

• Firstly, express the trigonometric function in terms of sine and cosine. This is typically the most straightforward starting point for simplification.


• Then apply known identities, like double angle or Pythagorean identities, to rewrite the expression. The goal here is to reach a form that can easily be compared or transformed into another equivalent expression.


• Simplify the resulting fractions or products to see if it matches the expected identity.

In the exercise provided, each expression was carefully rewritten using identities like \( \cos^2 u - \sin^2 u = (\cos u + \sin u)(\cos u - \sin u) \) and then simplified step by step. Such systematic approaches not only help in verifying identities but also build a deeper understanding of how trigonometric functions interact.