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Chapter 5: Problem 118

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 form of the given expression is 0.

Step by step solution

01

Apply the Logarithm Property

The logarithm property \(\ln (a) + \ln (b) = \ln (a*b)\) can be applied. Using this property, the given expression becomes: \(\ln (\cos^2t * (1+ \tan^2t)) \)
02

Identify the Trigonometric Identity

In the argument of the logarithm, identify the Pythagorean identity in trigonometry, \( \cos^2t + \sin^2t = 1 \) and, \(\tan^2t +1 = \sec^2t.\) Replace \(1+ \tan^2t\) by \(\sec^2t\). This results in: \(\ln (\cos^2t * \sec^2t)\)
03

Substitute Trigonometric Identity

Since \(\sec{t} = 1/ \cos{t}\), we find that \(\sec^2t = 1/ \cos^2t\). Substitute this into the equation from step 2 to get: \(\ \ln (\cos^2t / \cos^2t)\)
04

Perform Division

Perform the division to simplify the expression further to get: \(\ln (1)\)
05

Simplify

\(\ln (1) = 0\). This is the final result.

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