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Chapter 5: Problem 64

Rewrite the expression as a single logarithm and simplify the result. $$\ln \left(\cos ^{2} t\right)+\ln \left(1+\tan ^{2} t\right)$$

Short Answer

Expert verified
The simplified expression of the given logarithmic expression is 0

Step by step solution

01

Apply logarithm properties

The sum of two logarithms is equal to the logarithm of their multiplication. Using this rule, it can be rewritten as: \(\ln \left(\cos ^{2} t \times (1+\tan ^{2} t)\right)\)
02

Apply trigonometric identity

Based on trigonometric identity \(1+\tan^{2} t=\sec^{2} t\), the expression can be simplified as follows: \(\ln \left(\cos^{2} t \times \sec^{2} t\right)\)
03

Use the relationship between cosine and secant

Remember that secant is the inverse of cosine, and \( \sec^{2} t = 1/{(\cos^{2} t)} \). When multiplied, they give 1. As such, we simplify the expression to: \(\ln 1\)
04

Value of logarithm of 1

For any base, the logarithm of 1 is 0. Thus, the expression simplifies to: 0

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