Question

How do I prove the following Identity?

sec x(sec x-cos x)=tan^2x

If there is a certain website or suggestion to help with these type of equations I would greatly appreciate it. I've been studying these for a while but still get pretty confused.

Answer 1

sec(x)*[sec(x)-cos(x)]=

sec(x)*sec(x)-sec(x)*cos(x)

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

sec(x)*sec(x)-sec(x)*cos(x)=

1/cos(x) * 1/cos(x) - 1/cos(x) * cos(x)=

1/cos^2(x)-1=

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

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

Remark:

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

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

1-cos^2(x)/cos^2(x) = sin^2(x)/cos^2(x) = tan^2(x)

Answer 2

LS = sec x(sec x-cos x)
= (1/cosx)(1/cosx - cosx)
= 1/cos^2x - 1
= sec^2 - 1
= tan^2x (by definition)
= RS