Problem 58 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\cot x-\tan x}{\cot x+\tan x}=\cos 2 x $$

Step-by-Step Solution

Answer

Identity verified; both sides equal \( \cos 2x \).

Step 1: Write the Expressions in Terms of Sine and Cosine

Recall the definitions: \( \cot x = \frac{\cos x}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these into the left-side expression of the identity: \( \frac{\cot x - \tan x}{\cot x + \tan x} = \frac{\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}}{\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}} \).

Step 2: Obtain a Common Denominator

To simplify both the numerator and the denominator, find a common denominator for each. The common denominator for both is \( \sin x \cos x \). Hence, \( \frac{\cot x - \tan x}{\cot x + \tan x} = \frac{\frac{\cos^2 x - \sin^2 x}{\sin x \cos x}}{\frac{\cos^2 x + \sin^2 x}{\sin x \cos x}} \).

Step 3: Simplify the Fraction

The denominators \( \sin x \cos x \) cancel out in both the numerator and the denominator. This results in: \( \frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x} \).

Step 4: Use a Known Trigonometric Identity

Recall that \( \cos^2 x - \sin^2 x = \cos 2x \) and \( \cos^2 x + \sin^2 x = 1 \). Substitute these identities into the simplified expression: \( \frac{\cos 2x}{1} = \cos 2x \).

Step 5: Verify the Identity

We have shown that the left side of the given identity simplifies to \( \cos 2x \), which is exactly the right side of the identity, therefore proving the identity is verified.

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot x \), is one of the core trigonometric functions but is often less understood than its counterparts like sine or cosine. It is defined as the reciprocal of the tangent function: \[ \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \] Cotangent is especially useful in simplifying trigonometric expressions and identities. For example, in the trigonometric identity verification exercise, substituting \( \cot x \) with \( \frac{\cos x}{\sin x} \) helps to express the entire equation in terms of sine and cosine, which are easier to handle mathematically.

• Cotangent is undefined when \( \sin x = 0 \), as this makes the denominator zero.
• It has a periodicity of \( \pi \), repeating its values every \( \pi \) radians.

Understanding cotangent in relation to tangent helps in solving various trigonometric equations and identities.

Tangent

The tangent function, represented as \( \tan x \), is fundamental in trigonometry. It relates the sine and cosine functions in the following way: \[ \tan x = \frac{\sin x}{\cos x} \] In the context of verifying trigonometric identities, it's important to express \( \tan x \) in terms of \( \sin x \) and \( \cos x \) because it simplifies the task of comparing different parts of an equation. Specifically, when \( \tan x \) is involved in a complex expression, substituting with \( \sin x / \cos x \) reveals relationships that might not be immediately obvious.

• Tangent is undefined for angles where \( \cos x = 0 \), like \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc.
• It has a periodicity of \( \pi \), meaning the function repeats every \( \pi \) radians.

Recognizing when to use the definition of tangent in terms of sine and cosine is crucial for tackling and simplifying many trigonometric expressions.

Cosine

The cosine function, denoted as \( \cos x \), is integral to trigonometric identities and calculations. It is the x-coordinate of the point on the unit circle corresponding to an angle \( x \). In trigonometric expressions, cosine is often paired with sine. For identity verification exercises, like the one we analyzed, rewriting expressions using cosine can simplify and solve them. It works well in conjunction with other identities, such as:

• \( \cos^2 x + \sin^2 x = 1 \) is known as the Pythagorean Identity.
• Using representations like \( \cos x = \sqrt{1 - \sin^2 x} \) when necessary.

Cosine's behavior is cyclic, with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) radians.

Sine

The sine function, represented as \( \sin x \), is one of the most recognizable trigonometric functions. Similar to its counterpart, cosine, sine measures the y-coordinate of a point on the unit circle. It plays a crucial role in defining cotangent and tangent, as these two are expressed in terms of sine:

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

In trigonometry, sine is often combined with cosine to create identities and solve expressions. This function is periodic, completing its cycle every \( 2\pi \) radians, similar to cosine.
In verifying identities, expressing complex trigonometric terms into sine (and cosine) first makes them more manageable. This simplification technique was essential in solving the identity \( \frac{\cot x - \tan x}{\cot x + \tan x} = \cos 2x \).

Double Angle Formulas

The double angle formulas are vital tools in trigonometry used to simplify expressions involving angles that are double of a given angle. The formula for cosine in terms of a double angle is particularly useful: \[ \cos 2x = \cos^2 x - \sin^2 x \] This identity comes into play when you need to verify or prove trigonometric expressions, like turning \( \frac{\cot x - \tan x}{\cot x + \tan x} \) into a form involving \( \cos 2x \). By recognizing these known identities, complex expressions can often be reduced to simpler forms quickly.

• A second form of the cosine double angle formula is \( \cos 2x = 2\cos^2 x - 1 \).
• There are also double angle formulas for sine and tangent:
• \( \sin 2x = 2\sin x \cos x \)
• \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \) when \( \tan x eq 1 \).

Mastering these formulas is crucial for efficiently solving trigonometric problems and verifying identities.