91Ó°ÊÓ

The difference quotient is a key concept in calculus that helps us understand the rate of change of a function. When we take the function \(f(x)\) and alter it slightly to \(f(x+h)\), the difference in the function values over this small change in \(x\) gives us valuable information about the behavior of the function.

In this exercise, we're focusing on the function \(f(x) = \tan x\). To find the difference quotient, we use the formula:

• \(\frac{f(x+h) - f(x)}{h}\)

Our goal involves transforming this expression using trigonometric identities to eventually express it in a way involving \(\sec^2 x\). It is a fundamental step toward understanding derivatives in calculus, showing not only how functions behave locally but also how complex functions can be simplified.

Trigonometric identities are expressions that hold true for all values of the involved variables.

In this exercise, we extensively use the tangent addition formula:

• \(\tan(x + h) = \frac{\tan x + \tan h}{1 - \tan x \tan h}\)
• This formula is key to simplifying \(f(x+h) - f(x)\) when \(f(x) = \tan x\).

Trigonometric identities like \(1 + \tan^2 x = \sec^2 x\) also play a crucial role.

• Such identities transform expressions, aiding in simplification.

These transformations reduce complex trigonometric expressions into more manageable forms, which is essential in various calculus problems, enabling us to derive meaningful results more efficiently.

Limits are cornerstones of calculus that permit us to make sense of behaviors as an input tends to a certain value.

In problems like the one at hand, the limit concept is employed to match intermediate expressions to their most simplified forms. In our case:

• \(\lim_{h \to 0} \frac{\sin h}{h} = 1\)

Using this limit is crucial as \(h\) approaches zero, allowing us to effectively use the squeeze theorem to validate expressions and confirm these limiting behaviors accurately. This helps refine the difference quotient into the perfect form needed to connect with derivatives, showing just how vital limits are in endless calculus applications.

The tangent addition formula is fundamental in many trigonometry and calculus problems. It provides the ability to express the tangent of a sum of two angles with a known identity:

• \(\tan(x + h) = \frac{\tan x + \tan h}{1 - \tan x \tan h}\)

This formula, essential in the simplification steps of this exercise, lets us transform expressions involving \(\tan(x + h)\) and \(\tan x\) into a single fraction.

By substituting specific substitutions like \(\tan h = \frac{\sin h}{\cos h}\), we bridge our trigonometric identities with calculus notions. This showcase how important combining these identities is when tackling advanced problems involving limits and being able to navigate comfortably through is fundamental in finding neat and simplified end results.