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Chapter 5: Problem 63

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

Short Answer

Expert verified
\(\ln \left(\frac{\sec^2(t) - 1}{2\tan(t)}\right)\

Step by step solution

01

Combine the logarithms

Using the property of logarithms, the sum of two logarithms is equal to the logarithm of the product of their contents. This means that we can write \(\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)\) as \(\ln \left(|\cot t|\cdot \left(1+\tan ^{2} t\right)\right)\)
02

Use the Trigonometric Identity

The trigonometric identity \(1+\tan^2(t) = \sec^2(t)\) comes from the Pythagorean identity. Apply this identity to the expression to simplify it, we get - \(\ln \left(|\cot t|\cdot \sec ^{2} t\right)\)
03

Express cotangent in terms of secant

The cotangent is the reciprocal of the tangent and can be expressed in terms of secant as follows: \(\cot(t) = \frac{\cos(t)}{\sin(t)} = \frac{1}{\tan(t)} = \frac{\sec^2(t) - 1}{2\sec(t)\tan(t)}\). Substituting \(\cot(t) = \frac{\sec^2(t) - 1}{2\sec(t)\tan(t)}\) into the previous expression gives \(\ln \left(|\frac{\sec^2(t) - 1}{2\sec(t)\tan(t)}|\cdot \sec ^{2} t\right) = \ln \left(\frac{\sec^4(t) - \sec^2(t)}{2\sec(t)\tan(t)}\right)\)
04

Cancel out terms and simplify

We can cancel one term of \(\sec(t)\) from numerator and denominator. Also, using the fundamental trigonometric identity \(\sec(t) = \frac{1}{\cos(t)}\) and \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), we can simplify the expression to \(\ln \left(\frac{\sec^2(t) - 1}{2\tan(t)}\right)\) thus simplifying the expression as a single logarithm.

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Most popular questions from this chapter

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