Question

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Q no: 37
log(tan1)log(tan 2° )log(tan3° )...log(tan 45º
) =​

Answer 1

Step-by-step explanation:

What is log tan 1° + log tan 2° + …+log tan 89°?

Recall that

tan(90∘−x)=sin(90∘−x)cos(90∘−x)=cos(x)sin(x)=cot(x)

Therefore

log(tan(1∘))+log(tan(89∘))=log(tan(1∘)tan(89∘))=log(tan(1∘)cot(1∘))=log(1)=0

and similarly

log(tan(2∘))+log(tan(88∘))=0

log(tan(3∘))+log(tan(87∘))=0

⋮

log(tan(44∘))+log(tan(46∘))=0

and conveniently enough,

log(tan(45∘))=log(1)=0

so the whole sum is 0 . What’s really cool is the base of the logarithm doesn’t even matter.