Question

Please help me by solving this problem

Answer 1

We have :

tan1° tan2° tan3°........... tan87° tan88° tan89°

=> tan1° tan2°...........tan44° tan45° tan46°........tan88° tan89° .

=> ( tan1° * tan89° ) ( tan2° * tan88° )........ ( tan44° * tan46° ) * tan45°.

=> { tan1° * tan(90-1° ) } * { tan2° * tan(90-2°) }...... { tan44° * tan(90-44° ) } * tan45°.

=> ( tan1° * cot1° ) ( tan2° * cot2° )......... ( tan44° * cot44° ) × 1 [ Since Tan(90- theta) = Cot theta and Tan45° = 1 ]

=> ( 1 × 1 ×......×1 × 1 )

=> 1

Hence,

Tan1° Tan2° Tan3°.........Tan87° Tan88° Tan89° = 1....PROVED....

Answer 2

Given tan 1 tan 2 tan 3 .............. tan 87 tan 88 tan 89 .

It can be written as,

= > tan 1 tan 2 tan 3 .... tan 44 tan 45 tan 46 .... tan(90 - 3) tan(90 - 2) tan(90 - 1)

= > tan 1 tan 2 tan 3..... tan 44 (1) tan(90 - 44) ...... cot 3 cot 2 cot 1

= > tan 1 tan 2 tan 3...... tan 44 (1) cot 44 ...... cot 3 cot 2 cot 1

We know that cot theta = 1/tan theta

= > tan 1 tan 2 tan 3 ..... tan 44 tan (1) (1/tan 44)....... (1/tan 3) (1/tan 2) (1/tan 1).

= > (1).

Hope this helps!