Question

sec^4x-cos^4x=1-2cos^2x

Answer 1

Explanation:

We will use Pythagoras' identity:

sin2x+cos2x=1

from which we can deduce:

sin2x=1−cos2x

Also note that the difference of squares identity can be written:

A2−B2=(A−B)

We can use this with A=sin2x and B=cos2x as follows:

sin4x−cos4x=(sin2x)2−(cos2x)2

sin4x−cos4x=(sin2x−cos2x)(sin2x+cos2x)

sin4x−cos4x=sin2x−cos2x

sin4x−cos4x=(1−cos2x)−cos2x

sin4x−cos4x=1−2cos2x