Complete the proof of the identity by choosing the rule that justifies each step of

cos (x-y) / cosx cosy=1+tanx tany

To verify cos (x-y) / cosxcosy=1+tanxtany we proceed as follows:

cos (x+y) = cos x cos y-sin x sin y

considering left side we get

[cos (x+y) ]/[cos x cos y]

=[cos x cos y-sin xsin y]/[cos x cos y]

=1-[sin x sin y]/[cos x cosy]

=1-[sin x / cos x * sin y / cos y]

=1-tan x tan y

Hence the proof