Problem 8 - Algebra 2

Question

Verify that each of the following is an identity. $$ \cot x=\frac{\sin 2 x}{1-\cos 2 x} $$

Step-by-Step Solution

Answer

The identity \( \cot x = \frac{\sin 2x}{1 - \cos 2x} \) is verified.

Step 1: Understand the Identity Form

We are given the identity \( \cot x = \frac{\sin 2x}{1 - \cos 2x} \). The goal is to verify that this expression is valid by showing that both sides of the equation are equal for all \(x\) where both sides are defined.

Step 2: Express \( \cot x \) Using Trig Functions

Recall that \( \cot x = \frac{\cos x}{\sin x} \), which is the expression we will use for the left side of the identity.

Step 3: Simplify \( \frac{\sin 2x}{1 - \cos 2x} \)

For the right side, start by expressing \( \sin 2x \) and \( \cos 2x \) using their double-angle formulas. We have \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = \cos^2 x - \sin^2 x \).

Step 4: Substitute Double Angle Formulas

Substitute \( \sin 2x = 2 \sin x \cos x \) into \( \frac{\sin 2x}{1 - \cos 2x} \): \[ \frac{2 \sin x \cos x}{1 - (\cos^2 x - \sin^2 x)} \] \[ = \frac{2 \sin x \cos x}{1 - \cos^2 x + \sin^2 x} \].

Step 5: Simplify the Denominator

Since \( 1 - \cos^2 x = \sin^2 x \), substitute this in: \[ \frac{2 \sin x \cos x}{\sin^2 x + \sin^2 x} = \frac{2 \sin x \cos x}{2 \sin^2 x} \].

Step 6: Divide Each Term

Simplify by dividing both the numerator and the denominator by 2 \(\sin x\):\[ \frac{\sin x \cos x}{\sin^2 x} = \frac{\cos x}{\sin x} \].

Step 7: Conclude the Identity Verification

The right side \( \frac{\cos x}{\sin x} \) simplifies to \( \cot x \), which is the left side of the identity. Therefore, the initial equation is verified as an identity.

Key Concepts

Cotangent

The cotangent function, often denoted as \( \cot x \), is one of the primary trigonometric functions. It is defined as the reciprocal of the tangent function, which means it is equal to the ratio of the cosine of an angle to the sine of an angle. Mathematically, it can be expressed as:

• \( \cot x = \frac{\cos x}{\sin x} \)

Understanding cotangent is crucial since it frequently appears in various mathematical identities and equations. It's essential to keep in mind that \( \cot x \) is undefined whenever \( \sin x = 0 \), as division by zero is not possible.
Another interesting fact is that cotangent is closely related to tangent. You can express \( \cot x \) as \( \frac{1}{\tan x} \). This relationship highlights how trigonometric functions interconnect with one another, giving you different ways to handle problems requiring transformations or simplifications.

Double Angle Formulas

Double angle formulas are key tools in trigonometry that simplify the expressions of trigonometric functions. They are trigonometric identities that involve the sine, cosine, and tangent of an angle that is twice another angle. These formulas are invaluable for solving many types of trigonometric problems.

• The formula for sine is: \( \sin 2x = 2 \sin x \cos x \).
• The formula for cosine is: \( \cos 2x = \cos^2 x - \sin^2 x \), which can also be written in other forms such as \( 2\cos^2 x - 1 \) or \( 1 - 2\sin^2 x \).
• The formula for tangent is: \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \).

These formulas are used in various mathematical proofs and are especially useful when dealing with angles that are multiples of each other. Using double angle identities, you can transform complex trigonometric expressions into simpler ones, aiding in proving identities or solving equations.

Trigonometric Functions

Trigonometric functions are fundamental to trigonometry, a branch of mathematics that studies triangles and the relationships between their sides and angles. The six primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.Each function has a specific relationship with the angles and sides of a right triangle:

• Sine (\( \sin \theta \)): the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): the ratio of the opposite side to the adjacent side, or \( \frac{\sin \theta}{\cos \theta} \).
• Cotangent (\( \cot \theta \)): the reciprocal of the tangent, or \( \frac{\cos \theta}{\sin \theta} \).
• Secant (\( \sec \theta \)): the reciprocal of the cosine, or \( \frac{1}{\cos \theta} \).
• Cosecant (\( \csc \theta \)): the reciprocal of the sine, or \( \frac{1}{\sin \theta} \).

These functions are not just limited to angle measures but are also used for periodic phenomena like waves and circular motion. Understanding trigonometric functions and their properties allows students to tackle problems in both pure and applied mathematics.