Problem 95 - Algebra and Trigonometry

Question

Verify each identity. \(\ln e^{\tan ^{2} x-\sec ^{2} x}=-1\)

Step-by-Step Solution

Answer

After simplifying the given equation \(\ln e^{\tan^{2}x - \sec^{2}x} = -1\), both sides of the equation are equal. Therefore, the identity is verified.

Step 1: Apply the natural logarithm and exponential identity

Begin by applying the identity \(\ln e^x = x\) to the left side of the equation. Hence, \(\ln e^{\tan^2x - \sec^2x}\) simplifies to \(\tan^2x - \sec^2x\).

Step 2: Apply the trigonometric identity

Use the identity \(\tan^2x + 1 = \sec^2x\) which can be rewritten as \(\tan^2x - \sec^2x = -1\). Thus, the left-hand side simplifies to -1.

Step 3: Verification

The right hand side is -1. Now that both sides of the equation are equal, the given identity is verified.

Key Concepts

Natural Logarithm

In mathematics, the natural logarithm, denoted as \(\ln\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.718281828. The natural logarithm has many applications in calculus, particularly in solving equations involving exponential growth or decay.

One of the most important properties of the natural logarithm is the identity \(\ln e^x = x\). This property allows us to simplify complex expressions by transforming them into something more manageable. For example, if we have an expression like \(\ln e^{\tan^2 x - \sec^2 x}\), applying \(\ln e^x = x\) directly simplifies the expression to \(\tan^2 x - \sec^2 x\).

When working with natural logarithms, it's useful to remember that they undo the exponential function. This fact is pivotal in solving equations where exponential terms are present. It helps to transform problems into a simpler algebraic form, which is easier to manipulate.

Exponential Identity

The exponential identity is a fundamental concept when dealing with exponential functions. It helps simplify expressions that contain exponential terms.

A core principle of exponential identities is that the expression \(e^x\), when followed by a natural logarithm, returns the power itself (like \(\ln e^x = x\)). This identity is very useful in calculus for transforming expressions, especially when dealing with growth processes, decay, and when calculating certain limits.

Another useful identity involving exponential functions is \(e^{a+b} = e^a \cdot e^b\). This is the exponential sum identity and can be applied to simplify expressions involving sums in the exponent. Breaking down complex exponential terms using these identities transforms intricate expressions into simpler forms, making calculations more straightforward.

Exponential identities can be seamlessly combined with logarithmic identities to verify equations and complex mathematical expressions, as demonstrated in simplifying the given exercise.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using trigonometric identities. These identities relate various trigonometric functions together and are key in breaking down complex trigonometric problems.

A common identity used for simplification is \(\tan^2x + 1 = \sec^2x\). By rearranging this identity, we find \(\tan^2x - \sec^2x = -1\). This rearrangement is particularly useful for matching expressions and verifying identities, such as in our given problem.

Simplifying trig expressions also entails using algebraic manipulation combined with identities. Reducing trigonometric expressions to simpler forms can greatly aid in solving equations and proving identities.

• Recognize identity patterns quickly.
• Use substitution for complicated values.
• Combine algebraic manipulation with trig identities for optimal simplification.

Understanding these principles and techniques will very much smooth the path in working with trigonometric functions and identities in various mathematical problems.