Prove that the equation is an identity.
sec x/(sec x -tan x)=sec^2 x +sec x tan x
multiply top and bottom by sec+tan and recall that
sec^2-tan^2 = 1
To prove that the equation is an identity, we need to simplify both sides of the equation separately and show that they are equal for all values of x. Let's start by simplifying the left-hand side (LHS) of the equation:
LHS: sec x / (sec x - tan x)
To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator (sec x + tan x). Multiplying by the conjugate allows us to eliminate the denominators:
LHS: (sec x / (sec x - tan x)) * (sec x + tan x)
Using the distributive property, we can now simplify the numerator:
LHS: sec^2 x + sec x tan x
As you can see, we have obtained the same expression as the right-hand side (RHS) of the equation. Therefore, we can conclude that the equation is indeed an identity.
In summary, to prove that the equation is an identity, we simplified both sides of the equation and showed that they are equal using algebraic manipulations.