Problem 117 - Precalculus with Limits

Question

In Exercises 113 - 118, rewrite the expression as a single logarithm and simplify the result. \( \ln\mid\cot t\mid + \ln\left(1 + \tan^2 t\right) \)

Step-by-Step Solution

Answer

The final simplified expression of the given problem is \( \ln\left(\mid\frac{2}{\sin(2t)}\mid\right) \)

Step 1: Combine the logarithms

When two logarithms with the same base are added, they can be combined into one log where the arguments of the individual logs are multiplied. In this case, the following operation is done: \( \ln\mid\cot t\mid + \ln\left(1 + \tan^2 t\right) = \ln((\mid\cot t\mid) \cdot (1 + \tan^2 t)) \).

Step 2: Simplify the trigonometric component

We should note that 1 + \tan^2(y) is equal to \sec^2(y) in trigonometry. This simplifies the expression somewhat: \(\ln((\mid\cot t\mid) \cdot \sec^2 t) \)

Step 3: Further simplification

One might note that \sec(y) is equivalent to 1/\cos(y), and cot(y) is equivalent to \cos(y)/\sin(y). Thus, the equation can be rewritten as: \(\ln\left(\frac{\mid\cos(t)\mid}{\sin(t)} \cdot \frac{1}{\cos^2(t)}\right)\). By simplifying, the \(\cos(t)\) in numerator and denominator cancels out and you end up with \(\ln\left(\frac{\mid1\mid}{\sin(t) \cdot \cos(t)}\right)\). This simplifies to: \(\ln\left(\frac{1}{\sin(t) \cdot \cos(t)}\right)\)

Step 4: Final Simplification

Realizing that 2\sin(y)\cos(y) equals \sin(2y), and therefore \sin(y)\cos(y) is \(\frac{\sin(2y)}{2}\), it's possible to substitute this into our expression, to eventually end up with: \( \ln\left(\frac{2}{\sin(2t)}\right) \). As this is an absolute term, the final expression becomes: \( \ln\left(\mid\frac{2}{\sin(2t)}\mid\right) \)

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cotangent (\(\cot\)), secant (\(\sec\)), and cosecant (\(\csc\)). In the original exercise, we encounter two essential identities used to simplify expressions. These are:

• \(1 + \tan^2 t = \sec^2 t\)
• \(\cot t = \frac{\cos t}{\sin t}\)

By recognizing these identities, we transform complex expressions into simpler ones through substitution. Knowing when and how to use these identities helps in understanding and simplifying trigonometric parts of logarithmic expressions.

Logarithmic Properties

Logarithms have several useful properties, especially important here is the property of addition:

• When you add two logarithms with the same base, the result is the logarithm of the product of their arguments. In our case: \( \ln|\cot t| + \ln(1 + \tan^2 t) = \ln\left((|\cot t|) \cdot (1 + \tan^2 t)\right)\)

This property simplifies calculations by allowing us to deal with one logarithmic expression instead of two. Understanding these properties is crucial for effectively simplifying logarithmic expressions and turning them into more manageable forms.

Simplifying Expressions

Simplifying expressions, especially under the umbrella of logarithmic operations and trigonometric functions, is about applying step-by-step transformations to make a complex expression simpler or more concise. For the exercise we considered, the primary steps were:

• Combining the initial logs using properties of logarithms.
• Applying trigonometric identities to simplify within the logarithms.
• Performing algebraic manipulations to further condense the expression.

Finally, using trigonometric identities further, we get \(\ln\left(\left|\frac{2}{\sin(2t)}\right|\right)\). By substituting expressions for complex trigonometric values and simplifying their multiplicatives, you arrive at a much clearer and ideally, final form. By following these steps, you gain a refined and more intuitive grasp of manipulating and simplifying logarithmic and trigonometric expressions together.