Simplify the following. Show each step: (sin(x)/cos(x))² - 1/cos²(x)

The first step when subtracting two fractions is to check
if they have common denominator.

In this case, they both
have the denominator (cos x)^2, therefore, we can re-write the difference in this
way:

[(sin x)^2 - 1]/(cos
x)^2

We can use the Pythagorean identity, to replace
1:

(sin x)^2 + (cos x)^2 =
1

We'll re-write the
fraction:

[(sin x)^2 - (sin x)^2 - (cos x)^2]/(cos
x)^2

We'll eliminate like terms within the
brackets:

[- (cos x)^2]/(cos x)^2 =
-1

The result of subtracting the given
fractions is: (sin x)^2/(cos x)^2 - 1/(cos x)^2 =
-1.