[sec^2(x)sin^2(x)] / [sec(x) – cos(x)] = ?

First of all, I’d like to remove a “sec(x)” term from the numerator and denominator. To do this, I have to first factor out a “sec(x)” term from the denominator. Recall: sec(x) = 1/cos(x). And to factor out sec(x) from the denominator, I’ll have to divide sec(x) by sec(x), and divide cos(x) by sec(x):

cos(x) / sec(x) = cos (x) / [1 / cos(x)] = cos(x) * (cos(x) = cos^2(x)

So:

Step 1:

[sec^2(x)sin^2(x)] / [sec(x) – cos(x)]

= [sec^2(x)sin^2(x)] / [sec(x)][1 – cos^2(x)]

One sec(x) from the numerator now cancels out with one sec(x) in the denominator, so:

Step 2:

[sec^2(x)sin^2(x)] / [sec(x)][1 – cos^2(x)]

= [sec(x)sin^2(x)] / [1 – cos^2(x)]

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

=> 1 – cos^2(x) = sin^2(x), so:

Step 3:

[sec(x)sin^2(x)] / [1 – cos^2(x)]

= [sec(x)sin^2(x)] / [sin^2(x)]

Now, one sin^2(x) in the numerator cancels out with another in the denominator, so:

Step 4:

[sec(x)sin^2(x)] / [sin^2(x)]

= sec(x) = 1/cos(x)

Therefore, [sec^2(x)sin^2(x)] / [sec(x) – cos(x)]

= sec(x) = 1/cos(x).