Problem 75 - Contemporary Precalculus

Question

Prove the identity. $$\log _{10}(\cot x)=-\log _{10}(\tan x)$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity: $$\log _{10}(\cot x) = -\log _{10}(\tan x)$$ Answer: To prove this identity, we demonstrated that $$\log _{10}(\cot x) = \log _{10}(\frac{1}{\tan x}) = -\log _{10}(\tan x)$$. This involved expressing the cotangent function as the reciprocal of the tangent function and applying logarithmic properties.

Step 1: Recall the relationship between cotangent and tangent functions

The cotangent function, cot(x), can be expressed in terms of the tangent function, tan(x), as follows: $$\cot x = \frac{1}{\tan x}$$

Step 2: Apply the logarithm property of exponents

Using logarithm property of exponents, we get: $$\log _{10}(\frac{1}{\tan x}) = -\log _{10}(\tan x)$$

Step 3: Replace the cotangent function with the reciprocal of tangent function

Replace cot(x) with 1/tan(x) in the original equation: $$\log _{10}(\cot x) = \log _{10}(\frac{1}{\tan x})$$

Step 4: Compare the two logarithmic equations

We have now shown that: $$\log _{10}(\cot x) = \log _{10}(\frac{1}{\tan x}) = -\log _{10}(\tan x)$$ Thus, we have proved the identity: $$\log _{10}(\cot x) = -\log _{10}(\tan x)$$

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in understanding angles and their relationships. The primary trigonometric functions include sine, cosine, and tangent, each of which relate to the angles and sides of a right triangle.
One of the derivatives of these primary functions is the cotangent, denoted as \(\cot x\), which is the reciprocal of the tangent function. Knowing how these functions interconnect is crucial when solving mathematical identities or transformations.

• Sine (\(\sin x\)):\ Opposite side over hypotenuse.
• Cosine (\(\cos x\)):\ Adjacent side over hypotenuse.
• Tangent (\(\tan x\)):\ Opposite side over adjacent side, or \(\tan x = \frac{\sin x}{\cos x}\).
• Cotangent (\(\cot x\)):\ Reciprocal of tangent, or \(\cot x = \frac{1}{\tan x}\).

Understanding these relationships aids in grasping more complex mathematical problems, such as identities, which are equations that are true for every value of the variables contained within them.

Cotangent and Tangent Relationship

The relationship between cotangent and tangent is pivotal in trigonometry. Cotangent is essentially the flipped (or inverted) version of the tangent. This means when you have the tangent value of an angle, you can find the cotangent simply by taking its reciprocal, which means to "flip" the value. In mathematical terms, this is written as:

• \(\cot x = \frac{1}{\tan x}\)

This relationship is particularly helpful in solving identities, as it allows us to transform complex equations into simpler ones. In the context of logarithmic identities, understanding this reciprocal relationship is key because it allows us to use logarithm properties effectively. When you know one of these trigonometric values, you can easily find the other. This reciprocal relationship simplifies computations and helps in proving various trigonometric identities.

Logarithm Properties

Logarithms simplify the process of working with exponents and are widely used in mathematics, particularly in equations where variables are exponents. There are several key properties of logarithms that make them powerful tools:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^k) = k \cdot \log_b(M)\)

In the identity \(\log_{10}(\cot x) = -\log_{10}(\tan x)\), the Logarithm Quotient Rule is applied. By recognizing \(\cot x = \frac{1}{\tan x}\), the quotient rule simplifies the problem: \(\log_{10}(\frac{1}{\tan x}) = \log_{10}(\cot x)\) which equals \(-\log_{10}(\tan x)\). This showcases how understanding and leveraging logarithmic properties can aid in simplifying and proving complex mathematical expressions.