(secB-1)(secB+1)=tan^2B

L.H.S. (secB-1)(secB+1)

Put secB=1/cosB

[(1/cosB) - 1][(1/cos B) +1]
solving the brackets by multiplying the terms,we get

= (1/cosB)(1/cosB) + (1/cosB).1 + (1/cosB).(-1) + (1)(-1)
= 1/cos^2(B) + (1/cosB) + (-1/cosB) -1 (1/cosB and -1/cosB get cancelled)
therefore
= 1/cos^2(B) -1
taking the L.C.M.
= [1-cos^2(B)]/cos^2(B)

now sin^2(B) + cos^2(B)=1, therefore
1-cos^2(B)=sin^2(B)
putting this value in our L.H.S. we get

= sin^2(B)/cos^2(B)
= tan^2(B)
=R.H.S.

I hope it helped.