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How do you prove #(tan^3x - 1) /( tanx - 1) = tan^2x + tanx + 1#?

Answer 1

Madelyn Adams

as directed below

Please factorize numerator following the formula #a^3-b^3=(a-b)(a^2+ab+b^2)#. you will get result.

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Answer 2

Emery Adkins

To prove ( \frac{{\tan^3 x - 1}}{{\tan x - 1}} = \tan^2 x + \tan x + 1 ), we can follow these steps:

Rewrite the numerator ( \tan^3 x - 1 ) as ( (\tan^2 x + \tan x + 1)(\tan x - 1) ) using polynomial division.
Simplify the expression by canceling out common terms.
Verify that the resulting expression is equal to ( \tan^2 x + \tan x + 1 ).

Performing these steps will demonstrate the equality between the given expression and ( \tan^2 x + \tan x + 1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.