First, you should multiply the expression and simplify as far as you can. Then, write everything in terms of sine and cosine.
Here are the identities we'll use:
#color(white){color(black)(
(sectheta=1/costheta, qquadqquad(1.1)),
(sec^2theta=1/cos^2theta, qquadqquad(1.2)),
(tantheta=sintheta/costheta, qquadqquad(2.1)),
(tan^2theta=sin^2theta/cos^2theta, qquadqquad(2.2)),
(sin^2theta+cos^2theta=1, qquadqquad(3.1)),
(sin^2theta/cos^2theta+cos^2theta/cos^2theta=1/cos^2theta, qquadqquad(3.2)),
(tan^2theta+1=sec^2theta, qquadqquad(3.3)):}#
Some notes: identity #(1.2)# was achieved by squaring both sides of identity #(1.1)# (same with #(2.2)# and #(2.1)#).
Similarly, identity #(3.2)# was achieved by dividing all the terms in identity #(3.1)# by #cos^2theta#. Then, identity #(3.3)# was reached by simplifying identity #(3.2)# using previously-proved identities #(1.2)# and #(2.2)#
Now, here's the expression:
#color(white){color(black)(
((sectheta-1)(sectheta+1), qquadqquad"The problem"),
(sec^2theta+sectheta-sectheta-1, qquadqquad"Multiplying out the expression"),
(sec^2thetacolor(red)cancelcolor(black)(+sectheta-sectheta)-1, qquadqquad"Like terms cancel out"),
(sec^2theta-1, qquadqquad"Rewrite the above step"),
(tan^2theta+1-1, qquadqquad"Replace "sec^2theta" with "tan^2theta+1" using identity "(3.3)),
(tan^2thetacolor(red)cancelcolor(black)(+1-1), qquadqquad"Like terms cancel out"),
(tan^2theta, qquadqquad "Rewrite the above step"),
(sin^2theta/cos^2theta, qquadqquad "Replace "tan^2theta" with "sin^2theta/cos^2theta" using identity "(2.2)):}#
That's the answer. Hope this helped!
