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Chapter 5: Problem 61

Rewrite the expression as a single logarithm and simplify the result. $$\ln |\sin x|+\ln |\cot x|$$

Short Answer

Expert verified
The expression simplifies to \( \ln |\cos x|\)

Step by step solution

01

Combine the Logs

Combine the logarithms using the property \(\ln a + \ln b = \ln (ab)\). So \(\ln |\sin x| + \ln |\cot x| = \ln (|\sin x| |\cot x|)\).
02

Simplify the Expression

Simplify the expression: \(|\sin x|\) times \(|\cot x|\) = \(|\sin x| |1/\tan x|\) = \(|\sin x| / |\sin x / \cos x|\).
03

Eliminate the Same Terms

The \(|\sin x|\) in the numerator and the \(|\sin x|\) in the denominator will cancel when divided, leading to \(\ln (1 / |1 / \cos x|\) = \( \ln |\cos x|\).

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