Problem 80 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\cot 2 x=\frac{1-\tan ^{2} x}{2 \tan x}$$

Step-by-Step Solution

Answer

The identity \(\cot 2x = \frac{1-\tan^2 x}{2\tan x}\) is proven using trigonometric identities and simplification.

Step 1: Express Cotangent with Sine and Cosine

Recall that the cotangent function is the reciprocal of the tangent function, so \(\cot 2x = \frac{\cos 2x}{\sin 2x}\). We will use this identity for the left-hand side of the equation.

Step 2: Use Double Angle Identities

Utilize the double angle identities: \(\cos 2x = \cos^2 x - \sin^2 x\) and \(\sin 2x = 2 \sin x \cos x\). Substitute these identities into the expression for \( \cot 2x \):\[\cot 2x = \frac{\cos^2 x - \sin^2 x}{2 \sin x \cos x}\]

Step 3: Express With Tangent

Recall the identity \(\tan x = \frac{\sin x}{\cos x}\), which implies \(\sin x = \tan x \cos x\). Substitute \(\sin x = \tan x \cos x\) into the expression:\[ \cot 2x = \frac{\cos^2 x - \tan^2 x \cos^2 x}{2 \tan x \cos^2 x} \]

Step 4: Simplify the Expression

Factor out \(\cos^2 x\) from the numerator:\[\cot 2x = \frac{\cos^2 x (1 - \tan^2 x)}{2 \tan x \cos^2 x} \]

Step 5: Cancel Common Terms

Cancel \(\cos^2 x\) from the numerator and denominator:\[\cot 2x = \frac{1 - \tan^2 x}{2 \tan x} \]

Step 6: Conclude the Identity

The expression \(\frac{1 - \tan^2 x}{2 \tan x}\) on the right matches the original right-hand side provided in the identity. Thus, \(\cot 2x = \frac{1 - \tan^2 x}{2 \tan x}\) is proved.

Key Concepts

double angle identities

Double angle identities are a set of trigonometric formulas that express trigonometric functions of double angles in terms of single angles. They are particularly helpful in simplifying expressions and solving trigonometric equations.
For the cosine and sine functions, the double angle identities are:

• \( \cos 2x = \cos^2 x - \sin^2 x \)
• \( \sin 2x = 2 \sin x \cos x \)

These identities often prove useful when dealing with problems that require expressing complex trigonometric functions in simpler forms.
By applying these, it is easier to convert the trigonometric equation from being dependent on a double angle to single angles, which you directly used in Step 2 of your solution.

cotangent function

Cotangent, denoted as \( \cot x \), is one of the basic trigonometric functions. It is the reciprocal of the tangent function, meaning \( \cot x = \frac{1}{\tan x} \).
In terms of sine and cosine, the cotangent function can be expressed as \( \cot x = \frac{\cos x}{\sin x} \). This alternate form can be extremely useful when you're handling equations involving trigonometric identities or transformations, like in your exercise.
In the given solution, the expression for \( \cot 2x \) was written as \( \frac{\cos 2x}{\sin 2x} \), which leveraged the double angle identities to express \( \cos 2x \) and \( \sin 2x \) in terms of \( x \). This transformation was key to matching both sides of the trigonometric identity you're proving.

tangent function

The tangent function, symbolized as \( \tan x \), plays a fundamental role in trigonometry. It is defined as the ratio of the sine function to the cosine function: \( \tan x = \frac{\sin x}{\cos x} \).
This ratio highlights how the tangent function behaves and its dependencies on sine and cosine. Because tangent is directly related to sine and cosine, adjustments or transformations involving \( \tan x \) often require those two functions.
An essential property of \( \tan x \) is that it becomes undefined whenever \( \cos x = 0 \) because division by zero is impossible in mathematics. In your exercise, expressing \( \sin x \) in terms of \( \tan x \) (\( \sin x = \tan x \cos x \)) was critical for rewriting \( \cot 2x \) as \( \frac{1 - \tan^2 x}{2 \tan x} \). This manipulation ensures that the original double angle identity holds true.