Question

log tan1*log tan2.........................*logtan89=?

Answer 1

Log(tan1.tan2.tan3).................tan89)
=Log(tan1.tan89.tan2.tan88.tan3.tan87..........tan44.tan46.tan45)
=Log{tan1.tan(90-1).tan2.tan(90-2).tan3.tan(90-3)...........tan44.tan(90-46).tan45
=Log(tan1.cot1 . tan2.cot2. tan3.cot3 .........tan44.cot44.tan45)
=Log(1*1*1**.......1*1)
=log1 = 0

Answer 2

Answer : 0

Step by step explanation :

We look for special term, while solving such questions. In this case, neither tan0 nor tan90 are present. So, We will solve using tan45.

$= log( tan1 )\times log (tan2)... \times log(tan89) \\ = log( tan1 )\times log (tan2)....log(tan45).... \times log(tan89) \\ = log( tan1 )\times log (tan2)....log1.... \times log(tan89) \\ = log( tan1 )\times log (tan2)....... \times 0 \times log(tan89) \\= 0$

Therefore, We used two properties

1. Zero multiplied to anything would be zero.

2. Log1 = 0