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Chapter 4: Problem 2

Explain why \(\cot x \cdot \tan x=1\)

Short Answer

Expert verified
\(\cot x \cdot \tan x = 1\) because when the definitions \(\cot x = \frac{\cos x}{\sin x}\) and \(\tan x = \frac{\sin x}{\cos x}\) are applied, the expression simplifies to 1.

Step by step solution

01

Understand the Trigonometric Identities

Recognize that \(\cot x\) is the cotangent of \(x\) which is defined as \(\frac{1}{\tan x}\) or \(\frac{\cos x}{\sin x}\), and \(\tan x\) is the tangent of \(x\) which is defined as \(\frac{\sin x}{\cos x}\).
02

Simplify the Expression

Simplify \(\cot x \times \tan x\) by substituting in the definitions: \[\cot x \times \tan x = \left(\frac{\cos x}{\sin x}\right) \times \left(\frac{\sin x}{\cos x}\right)\].
03

Cancel Common Factors

Cancel out the common factors in the numerator and denominator: \[\left(\frac{\cos x}{\sin x}\right) \times \left(\frac{\sin x}{\cos x}\right) = \frac{\cos x \times \sin x}{\sin x \times \cos x}\].
04

Conclude with the Result

Observe that all terms cancel each other out, leaving \(\frac{\cos x \times \sin x}{\sin x \times \cos x} = 1\), proving that \(\cot x \times \tan x = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent
Understanding the cotangent function is crucial when dealing with trigonometric identities. It is one of the six fundamental trigonometric functions and is commonly abbreviated as 'cot'. The cotangent of an angle in a right-angled triangle is the ratio of the adjacent side to the opposite side. In more general terms for any angle \( x \), the cotangent is the reciprocal of the tangent function, written as \( \text{cot}(x) = \frac{1}{\tan(x)} \textrm{or}\ \frac{\text{cos}(x)}{\text{sin}(x)} \). This reciprocal relationship forms the basis of one of the trigonometric identities we are exploring.
Tangent
The tangent function, often denoted as 'tan', is also one of the primary trigonometric functions. In contrast to cotangent, for any given angle \( x \), tangent represents the ratio of the side opposite to the angle to the adjacent side in a right-angled triangle. Algebraically, it is defined as \( \tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \textrm{or simply}\ \frac{\text{opposite}}{\text{adjacent}} \). Tangent and cotangent are closely related and understanding both functions helps in grasping their role in trigonometric identities.
Simplifying Expressions
Simplifying expressions is a fundamental skill in mathematics, particularly when working with trigonometric identities. The goal is to transform complex expressions into simpler or more standard forms. This process often involves substituting defined expressions, factoring, expanding, and canceling like terms. In trigonometry, simplifying may require using fundamental identities to find more concise or equivalent expressions. For instance, multiplying cotangent and tangent, and recognizing that their product simplifies to one is an excellent example of simplification.
Trigonometric Functions
Trigonometric functions are a cornerstone of geometry and calculus. They relate the angles of a triangle to the lengths of its sides and have a wide range of applications including in science, engineering, and navigation. Besides cotangent and tangent, the other core trigonometric functions are sine (sin), cosine (cos), secant (sec), and cosecant (csc). Each function has a specific relationship with the others, often depicted in the form of identities. These identities are useful for simplifying complex expressions and solving equations.

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Most popular questions from this chapter

Calculate the exact value of the inverse function geometrically. Assume the principal branch in all cases. Check your answers by direct calculation. $$\tan \left(\cot ^{-1} 4\right)(\text { Surprise? })$$

Show the steps in trans forming the expression on the left to the one on the right. $$\begin{aligned} &(\sec E-1)(\sec E+1)\\\ &\text { to } \tan ^{2} E \end{aligned}$$

What is the reciprocal property for sec \(x ?\)

Calculate the exact value of the inverse function geometrically. Assume the principal branch in all cases. Check your answers by direct calculation. $$\cos \left(\sin ^{-1}\left(-\frac{8}{17}\right)\right)$$

If you enter \(\cos ^{2} 0.7\) and \(\sin ^{2} 0.7\) into your calculator, you get these numbers: $$ \begin{array}{l} \cos ^{2} 0.7=0.5849835715 \\ \sin ^{2} 0.7=0.4150164285 \end{array} $$ Without using your calculator, do the addition. What do you notice?
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