Problem 70 - Contemporary Precalculus

Question

Prove the identity. $$\frac{\sin x}{1-\cot x}+\frac{\cos x}{1-\tan x}=\cos x+\sin x$$

Step-by-Step Solution

Answer

Q: Prove the trigonometric identity: $$\frac{\sin x}{1-\cot x}+\frac{\cos x}{1-\tan x}=\cos x+\sin x$$ A: The given identity is incorrect and cannot be proven true, as shown through the step-by-step solution provided. The left side simplifies to 0, which is not equal to the right side, $\cos x + \sin x$.

Step 1: Rewrite the cotangent and tangent functions in terms of sine and cosine functions.

To do this, we will use the definitions of cotangent and tangent: $$\cot x = \frac{\cos x}{\sin x} \quad\text{and}\quad \tan x = \frac{\sin x}{\cos x}$$ Now rewrite the expression using these definitions: $$\frac{\sin x}{1-\frac{\cos x}{\sin x}}+\frac{\cos x}{1-\frac{\sin x}{\cos x}}$$

Step 2: Simplify the denominators.

To simplify, we will find a common denominator for the fractions in the denominators. First, multiply the numerators and denominators by the necessary factors. $$\frac{\sin x.\sin x}{\sin x-\cos x}+\frac{\cos x.\cos x}{\cos x-\sin x}$$

Step 3: Simplify the numerator and denominator, and factor out common terms.

Combine the two fractions by finding a common denominator and simplify the numerator and denominator. $$\frac{\sin^2 x - \cos^2 x}{(\sin x-\cos x)(\cos x-\sin x)}$$

Step 4: Use the Pythagorean identity and rearrange the terms.

Apply the Pythagorean identity \(\sin^2 x+\cos^2 x=1\). Rewrite the numerator using it and simplify the denominator. $$\frac{1 - (\sin^2 x+\cos^2 x)}{(\sin x-\cos x)(-(\sin x-\cos x))}$$

Step 5: Simplify the expression.

Now, simplify the numerator and cancel out the common terms in the numerator and denominator. $$\frac{1-1}{(\sin x-\cos x)^2} = \frac{0}{(\sin x-\cos x)^2} = 0$$ Since we got 0 on the left side, the identity is false. However, it is important to note this exercise is incorrectly formulated, as both sides of the equation are not equal. In other words, the given identity is not correct.

Key Concepts

Trigonometric Functions

Trigonometric functions form the backbone of trigonometry and are crucial in connecting angles with ratios of sides in right-angled triangles. The primary trigonometric functions are sine (\( \sin x \)) and cosine (\( \cos x \)). Each function provides vital information about angular measurements.

• **Sine**: Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• **Cosine**: Represents the ratio of the adjacent side to the hypotenuse.
• **Tangent** (\( \tan x \)): The ratio of sine to cosine, given by \( \tan x = \frac{\sin x}{\cos x} \).
• **Cotangent** (\( \cot x \)): The reciprocal of tangent, or \( \cot x = \frac{\cos x}{\sin x} \).

Different trigonometric functions allow us to express complex relationships such as angles and distances. They are broadly applied in fields like physics, engineering, and geometry. Understanding how these functions transform and simplify expressions is essential in solving trigonometric equations.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It represents a vital relationship between the sine and cosine functions. This identity states:\[\sin^2 x + \cos^2 x = 1\]This equation highlights that for any angle \(x\), the sum of the square of the sine and cosine functions equals one. This identity derives from the Pythagorean theorem applied in the unit circle context, where the radius equals one.Using the Pythagorean identity allows us to transform trigonometric expressions, simplify complex equations, and verify identities. For instance, if you know the value of \(\sin x\), you can easily find \(\cos x\) and vice-versa using this relation. It is a powerful analytical tool that underlies many other identities and solution methods in trigonometry.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and expressions to simplify, solve, or prove mathematical statements. In trigonometry, it often involves transforming expressions using identities like Pythagorean, reciprocal, or quotient identities. To manipulate trigonometric expressions effectively:

• Convert complex expressions into simpler forms by changing trigonometric functions using known identities.
• Clear fractions by multiplying the numerator and the denominator by common factors.
• Combine like terms and simplify fractions by factoring and canceling common terms.
• Apply strategic rearrangements and substitutions to reach the desired form.

Algebraic manipulation is critical when proving trigonometric identities and involves creative approaches to solving problems. The goal is to systematically transform and simplify expressions to validate equations or uncover underlying properties.