As in sum and difference formulas of sine and cosine, you use both of these ratios, for #sec(a+-b)#, one will need both secant and cosecant. But you can write cosecant again as secant, albeit in a bit complicated form.
#sec(a+b)=1/cos(a+b)=1/(cosacosb-sinasinb)#
= #1/(1/(secasecb)-1/(cscacscb)#
= #(secasecbcscacscb)/(cscacscb-secasecb)#
Similarly #sec(a-b)=(secasecbcscacscb)/(cscacscb+secasecb)#
Now #cscx=1/sinx=1/sqrt(1-cos^2x)=1/sqrt(1-1/sec^2x)#
= #1/sqrt((sec^2x-1)/sec^2x)=secx/sqrt(sec^2x-1)#
So you can replace cosecant accordingly and
#sec(a+b)=(secasecbseca/sqrt(sec^2a-1)secb/sqrt(sec^2b-1))/((seca/sqrt(sec^2a-1)secb/sqrt(sec^2b-1))-secasecb)#
and #sec(a-b)=(secasecbseca/sqrt(sec^2a-1)secb/sqrt(sec^2b-1))/((seca/sqrt(sec^2a-1)secb/sqrt(sec^2b-1))+secasecb)#
