Problem 73 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Prove the following identity $$\frac{\cos x \cot x}{\cot x-\cos x}=\frac{\cot x+\cos x}{\cos x \cot x}$$ Answer: To prove this identity, we first convert cot(x) into sin(x) and cos(x) using its definition \(\cot x = \frac{\cos x}{\sin x}\). Next, we simplify both sides of the equation and factor out common terms. Finally, we cancel out the common terms on both sides, which results in the original given identity: $$\frac{\cos x \cot x}{\cot x-\cos x}=\frac{\cot x+\cos x}{\cos x \cot x}$$

Step 1: Write down the given identity

The given identity is: $$\frac{\cos x \cot x}{\cot x-\cos x}=\frac{\cot x+\cos x}{\cos x \cot x}$$

Step 2: Convert cot(x) into sin(x) and cos(x)

Recall that \(\cot x = \frac{\cos x}{\sin x}\). We can substitute this definition of cot(x) into the given equation: $$\frac{\cos x \cdot \frac{\cos x}{\sin x}}{\frac{\cos x}{\sin x}-\cos x}=\frac{\frac{\cos x}{\sin x}+\cos x}{\cos x \cdot \frac{\cos x}{\sin x}}$$

Step 3: Simplify both the sides of the equation

Now, we will simplify both the sides of the equation. For the left side: $$\frac{\cos^2 x}{\sin x(\frac{\cos x}{\sin x}-\cos x)}$$ $$\frac{\cos^2 x}{\frac{\cos x-\cos x\sin x}{\sin x}}$$ $$\frac{\cos^2 x\sin x}{\cos x-\cos x\sin x}$$ For the right side: $$\frac{\frac{\cos x+\cos x\sin x}{\sin x}}{\cos^2 x}$$ $$\frac{(\cos x+\cos x\sin x)}{\sin x \cos^2 x}$$ Now, both sides of the equation look like: $$\frac{\cos^2 x\sin x}{\cos x-\cos x\sin x}=\frac{(\cos x+\cos x\sin x)}{\sin x \cos^2 x}$$

Step 4: Factor out common terms

We can factor out the common terms in both the numerator and denominator on both sides. For the left side: $$\frac{\cos x\sin x(\cos x)}{\cos x(1-\sin x)}$$ For the right side: $$\frac{\cos x (1+\sin x)}{\sin x \cos x(\cos x)}$$ Now, both sides of the equation look like: $$\frac{\cos x\sin x(\cos x)}{\cos x(1-\sin x)}=\frac{\cos x (1+\sin x)}{\sin x \cos x(\cos x)}$$

Step 5: Cancel out common terms

We can cancel out the common terms on both sides of the equation. $$\frac{\cancel{\cos x}\sin x(\cos x)}{\cancel{\cos x}(1-\sin x)}=\frac{\cancel{\cos x} (1+\sin x)}{\sin x \cancel{\cos x}(\cos x)}$$ $$\frac{\sin x(\cos x)}{(1-\sin x)}=\frac{ (1+\sin x)}{\sin x (\cos x)}$$ Now, we can observe that both sides of the equation are equal. Thus, we have proved the given identity: $$\frac{\cos x \cot x}{\cot x-\cos x}=\frac{\cot x+\cos x}{\cos x \cot x}$$

Key Concepts

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable within the domain. They are essential for simplifying expressions and solving equations involving trigonometric functions. An example of a basic trigonometric identity is the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), which relates the sine and cosine of an angle. In the exercise provided, the goal was to prove a more complex identity involving both cosine and cotangent functions by demonstrating that both sides of the equation are equivalent under all circumstances where they are defined.

To approach these problems, one typically starts by using known identities, such as the cotangent definition, to rewrite the equation in terms of simpler functions. After which, transforming both sides of the identity separately through algebraic manipulation and simplification leads to a point where they can be directly compared or shown to be identical. The ability to recognize and use trigonometric identities skillfully is central to success in solving such problems.

Cotangent Definition

In trigonometry, cotangent is one of the six fundamental trigonometric functions. Defined as the reciprocal of the tangent function, the cotangent of an angle \(x\) can be expressed as \(\cot x = \frac{1}{\tan x}\), or, in terms of sine and cosine, as \(\cot x = \frac{\cos x}{\sin x}\). Understanding this definition is vital because it allows us to transform the cotangent function into equivalent expressions involving sine and cosine, making it more manageable to work with in proofs, as was demonstrated in the second step of the step-by-step solution. In trigonometry and calculus, recognizing when and how to apply such definitions can simplify complex equations and make it possible to solve problems that initially appear daunting.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics that involves reducing equations to a more manageable form while keeping their values unchanged. The process often includes rewriting terms using algebraic properties, combining like terms, and canceling out common factors. In the context of the trigonometric identity problem, simplification was crucial in moving from a more complex expression to one where the underlying equivalence becomes clear. Simplification can also include the use of trigonometric identities to replace functions with equivalent expressions, as shown in the example problem where the cotangent function was replaced with its equivalent in terms of sine and cosine. By methodically simplifying each side of the equation, we draw closer to the comparison or eventual cancellation of terms, ultimately proving the identity.

Factoring Algebraic Expressions

Factoring is a powerful algebraic technique for simplifying expressions and solving equations. To factor an expression means to write it as a product of its factors—simpler expressions whose product is the original expression. In the trigonometric identity proof provided, factoring was used in step 4, when common terms in the numerator and denominator were factored out. This is similar to factoring a polynomial where we look for common terms or use identities to create them. Factoring aids in reducing complex expressions to simpler forms and often makes it possible to cancel out common terms, moving us closer to proving the identity. It's particularly useful in trigonometry, where expressions can become unwieldy, and recognizing common factors can be the key to simplification. Mastery of this technique is a cornerstone for anyone looking to excel in higher-level mathematics.