Problem 30 - Algebra and Trigonometry

Question

Verify each identity. $$ \cot x=\frac{1+\cos 2 x}{\sin 2 x} $$

Step-by-Step Solution

Answer

The identity \(\cot x=\frac{1+\cos 2 x}{\sin 2 x}\) is correct as verified by the step-by-step solution.

Step 1: Break down the left-hand side

Rewrite the left-hand side \(\cot x\) as \(\frac{\cos x}{\sin x}\) as this is the identity of \(\cot x\).

Step 2: Break down the right-hand side

Rewrite \(\cos 2x\) and \(\sin 2x\) using the double angle identity. \(\cos 2x\) becomes \(2\cos^2 x -1\) and \(\sin 2x\) becomes \(2\sin x \cos x\). Substitute these into the right-hand side equation.

Step 3: Simplify the right-hand side

Substitute the expressions for \(\cos 2x\) and \(\sin 2x\) into the right-hand side equation to get \(\frac{1 + 2\cos^2 x -1}{2\sin x \cos x}\). This simplifies to \(\frac{2\cos^2 x}{2\sin x \cos x}\), which further simplifies to \(\frac{\cos x}{\sin x}\) after cancelling common terms.

Step 4: Compare both sides

Both the left-hand side and the right-hand side are the same, \(\frac{\cos x}{\sin x}\). Therefore the identity is verified.

Key Concepts

Cotangent

The cotangent function is one of the basic trigonometric functions, often abbreviated as \( \cot x \). It is fundamentally related to both the sine and cosine functions. Here's how you can understand it:

• Definition: Cotangent is defined as the quotient of the cosine and sine of an angle. Hence, \( \cot x = \frac{\cos x}{\sin x} \).


• Reciprocal: It is the reciprocal of the tangent function, meaning \( \cot x = \frac{1}{\tan x} \).


• Role: Cotangent displays the ratio of the adjacent side to the opposite side in a right-angled triangle.

Understanding cotangent is vital because it simplifies into many expressions and is often used where simple divisions of sine and cosine are involved, like the identity verification in our exercise.

Double Angle Formulas

Double angle formulas are trigonometric identities that allow you to express trigonometric functions of doubled angles, such as \( 2x \), in terms of functions of single angles \( x \). These formulas are integral for simplifying and transforming trigonometric expressions. Here are the key ones:

• Double angle for cosine: \( \cos 2x = 2\cos^2 x - 1 \). Alternatively, it can also be expressed as \( \cos 2x = 1 - 2\sin^2 x \).


• Double angle for sine: \( \sin 2x = 2\sin x \cos x \).

These formulas are extremely helpful when working with identities because they make it possible to break down complex expressions into simpler ones. As you can see from our exercise, using these formulas allowed the transformation of \( \cos 2x \) and \( \sin 2x \) into more manageable terms.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using known identities and algebraic manipulations to transform complex expressions into simpler equivalents. Here's how it's most often approached:

• Trigonometric Identities: Use identities like the Pythagorean identity, reciprocal identities, and double angle formulas to simplify expressions.


• Factorization: Factor expressions when possible, to reveal hidden simplifications. In many cases, this involves recognizing common terms that can cancel each other out.


• Reduction: Replace more complex terms with simpler equivalents by reducing fractions or converting sums into products.

In the exercise, simplifying the expression \( \frac{1 + \cos 2 x}{\sin 2 x} \) by substituting double angle formulas and rearranging terms until a match with the cotangent definition was found forms a prime example of simplification at work. This ability to reduce and equate complex expressions, especially involving trigonometric functions, is crucial in verifying identities efficiently.