Question

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

Answer 1

Ans=>
           tan 1.tan 2. tan3. .... tan 45......tan 87.tan 88. tan89.
       =>tan 1.tan 2.tan 3.......1............tan(90-3).tan(90-2).tan(90-1).
       =>tan 1.tan 2.tan 3........1...........cot 3.cot 2.cot 1.
       =>1.(Ans.)  
    

Answer 2

Extending the to first 4 multiple and last 4 too:

tan1.tan2.tan3.tan4………tan86.tan87.tan88.tan89

Note convert first 4 tan to cot (by using trignomatrical ratio of complementary angle - tan(90-A)=cotA & cot(90-A)=tanA )

Now,

cot89.cot88.cot87.cot86…………tan86.tan87.tan88.tan89

We know that cotA×tanA=1

So,

cot89.tan89.cot88.tan88.cot87.tan87.cot86.tan86……………

1.1.1.1.………

Therefore, 1