Question

prove that tan1.tan2.tan3....................tan89=1

Answer 1

hope this answer will help you.

Answer 2

tan 1 . tan 2 . tan 3 ........ tan 89

We can write the above problem like this :

tan 1 tan 89 . tan 2 tan 88 . ..............................

Note that we are multiplying the last element with the first one and then we are moving onto the next element from the last .

tan 1 . tan 89 . tan 2 tan 88 ......

Now tan ( 90 - A ) = cot A

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

Hence tan 2 = tan ( 90 - 88 ) = cot 88 .

Like this every element will be replaced :

cot 89 . tan 89 . cot 88 . tan 88 ......................

We know that cot A tan A = 1

This means that every multiplication product will be equal to 1 :

1 . 1 . .................

= > 1

The final answer will be 1 because every element will be equal to 1 when arranged like that !

Hence Proved.