Math, asked by sahajvirk16, 25 days ago

do not use trigonometric tables. ​

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Answered by mfb8525
1

LHS = tan 1 * tan 2 * tan 3 * .. tan89

       = tan 1 * tan 2 * tan 3 * .. tan 45 *.. tan 87 * tan 88 * tan 89

       = cot (90 - 1)  * cot(90 - 2) * cot(90 - 3) *.. 1*.. tan 87 * tan 88 * tan 89

       = cot 89 cot 88 cot 87 * .. * tan 87 tan 88 tan 89

Now we know cot x * tan x = 1/tan x * tan x = 1, so all cot and tan will cancel out and we get,

      = 1

      = RHS

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Answered by pratimadevigee
1

Recall that

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

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

os⁡(x)sin⁡(x)=cot⁡(x)

Therefore

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

log(tan(1∘)cot(1∘))=log(1)=0log⁡(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∘))=0log⁡(tan⁡(2∘))+log⁡(tan⁡(88∘))=0

log(tan(

2 tanθ=cot(90∘−θ)

∴tan1∘=cot89∘

cot89∘=tan89∘1

Similarly all the term will cancel out each other, except tan45∘, whose value is equal to 1.

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