Question

Find the value of the expression tan1tan2.......tan89

Answer 1

tan1.tan2.........tan89


tan 89 = tan(90-1) =cot 1

[tan(90-A)=cotA]

likewise.....

tan 88 =cot 2
tan 87= cot 3
................up to tan 46=cot 44

middle one is tan 45 =1

so it is

tan1.tan2...........tan44.tan 45.cot44..............cot1



[tanA*cotA=1]


tan1.tan2...........tan44.tan45.cot44..............cot1 = 1

So,answer = 1

Hope it helps

Answer 2

Answer is 1 because
$\tan( \alpha) \tan(90 - \alpha ) = 1$
So everything will get cancelled out except tan45 which is equal to 1. Thanks.