Let #secalpha=a, secbeta=b and secgamma=c#.
Since, #AA theta in RR-{(2k+1)pi/2| k in ZZ}, |sectheta| >=1#, we find
that, #|a|>=1, |b|>=1, |c|>=1#.
#rArr tanalpha=+-sqrt(a^2-1), tanbeta=+-sqrt(b^2-1)#, and,
#tangamma=+-sqrt(c^2-1)# are defined as #|a|, |b|, |c| >=1#.
Then, by what is given, we have,
#a=bc+-sqrt((b^2-1)(c^2-1)) rArr(a-bc)^2=(b^2-1)(c^2-1)#
#rArr a^2-2abc+cancel(b^2c^2)=cancel(b^2c^2)-b^2-c^2+1#
#rArr b^2-2abc=1-a^2-c^2#
Adding #a^2c^2# on both sides, we have,
#b^2-2abc+a^2c^2=a^2c^2-a^2-c^2+1#
#rArr (b-ac)^2=a^2(c^2-1)-1(c^2-1)=(a^2-1)(c^2-1)#
#rArr b-ac=+-sqrt((a^2-1)(c^2-1))#
#rArr b=ac+-sqrt((a^2-1)(c^2-1)#
#rArr secbeta=secalphasecgamma+-tanalphatangamma#.
Hence, the Proof. Enjoy Maths.!