Question

The value of logtan1° + logtan2° +………..+logtan89° is :
1. -1
2. 0
3. 1
4. π /2 

Answer 1

Correct Answer - Option 2 : 0

Formula Used:

Trigonometric functions

tanx × cotx = 1

Tan(90°-x) = cotx

Property of logarithm 

loga + logb+ logc = log(abc)

Calculation:

Let y =  logtan1° + logtan2° +………..+logtan89°

⇒ y = logtan1° + logtan2° +…..+ logtan45° + logtan46° +…..+ logtan89°

⇒ y  = logtan1° + logtan2° +……+ logtan45°+ logcot(90°- 46°) +…….+ logcot(90°- 89°)

⇒ y = logtan1°+ logtan2°+………logtan44°+ log(1) + logcot44°+………+ logcot1°

From property of logarithm we have

loga + logb+ logc = log(abc)

⇒ y = log(tan1°.tan2°.tan3°……tan44°.1.cot44°…….cot1°)

We know from trigonometric functions

tanx × cotx=1

So can celling the consecutive terms in the above equation we get

⇒ y = log(1.1.1.. …1)

⇒ y = log(1)

⇒ y = 0

So,

∴ logtan1° + logtan2° +………….+ logtan89° = 0