91Ó°ÊÓ

The tangent function, often written as \(\tan(x)\), is one of the basic trigonometric functions. It relates an angle of a right-angled triangle to the ratio of the opposite side to the adjacent side.
In other words, if you know the lengths of the sides, you can find the angle, and vice versa.
Mathematically, it is defined as:
\(\tan(x) = \frac{\text{opposite}}{\text{adjacent}}\).
The tangent function is periodic with a period of \(\frac{\text{Ï€}}{\text{Ï€}}\) and it has vertical asymptotes at odd multiples of \(\frac{\text{Ï€}}{2}\). This means the function tends to infinity as it approaches these points.
Using the tangent function in different algebraic and trigonometric manipulations is fundamental in solving various mathematical problems.

The difference of squares is a useful algebraic identity that helps simplify expressions of the form \(a^2 - b^2\).
The formula for the difference of squares is:
\(a^2 - b^2 = (a-b)(a+b)\).
This identity allows us to break down complex products into simpler factors.
In the given exercise, the expression \((\tan x-1)(\tan x+1)\) is a difference of squares:
Here, you can identify \(\tan x\) as \(a\) and 1 as \(b\), thus:
\((\tan x-1)(\tan x+1) = (\tan^2 x - 1)\).
This transformation simplifies our work and makes it easier to combine terms later on.

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions using various algebraic identities and operations.
It helps in solving equations and simplifying complex expressions.
In our step-by-step solution:

• We started with the expression \(\tan x(\tan x-1)(\tan x+1)\).
• By recognizing the difference of squares identity, we rewrote \((\tan x-1)(\tan x+1)\) as \((\tan^2 x - 1)\).
• Finally, we multiplied \(\tan x\) with \(\tan^2 x - 1\), leading to \(\tan^3 x - \tan x\).

This final product is a more straightforward form of the initial expression, making it easier to work with in further calculations or evaluations.
Mastering algebraic manipulation involves constant practice and familiarity with various identities and operations.