Question

tan 1, tan2, tan3.......................tan88 ,tan 89

Answer 1

tan 1 . tan 2 .tan 3 . .......tan 88 . tan 89
=sin1/cos1 .sin 2/cos 2  .  ...sin 45/cos 45 .....sin 88/cos 88 . sin 89/ cos 89
=sin(90-89)/cos 1 . sin(90-88)/cos 2...........sin(90-45)/cos 45.........sin(90-1)/cos89
=cos 89/cos1 .cos 88/cos 2......cos 45/cos 45 ..........cos 1/ cos 89

all the values will get cancelled......and the result will be 1

Answer 2

The answer will be '1' .

Example:- 
Let's take tan 89 and tan 1 in the first case and Let's take Tan 2 and tan 88 in 2nd case:-
By the Problem,[1st case]:-
=) tan(90-89) tan89
=) cot89 x tan89
=) 1/tan89 x tan89  [tan89 will be canceled on both the sides by multiplying]
=) 1. Ans.

By the Problem,[2nd Case]:-
=) tan(90-88) tan88
=) cot88 x tan88
=) 1/tan88 x tan88 [tan 88 will be canceled on both the sides by multiplying]
=) 1.Ans.

Therefore, if we go on doing this for every ratios...we will come to the answer 1 for each case.
Hence, Answer = 1.