How do I show that (cos^4x - sin^4x) / cos^2x = 1 - tan^2x

Start with the LHS:

(cos^4x - sin^4x) / cos^2x 

Recognise the difference of two squares on the top line, which simplifies to (cos^2x - sin^2x)(cos^2x + sin^2x):

(cos^2x - sin^2x)(cos^2x + sin^2x) / cos^2x

Because of the identity sin^2x + cos^2x = 1, the second bracket (cos^2x + sin^2x) simplifies to 1:

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

Separate the two parts of the numerator:

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

These parts both simplify to 1 and tan^2x respectively:

1 - tan^2x 

= RHS