Question

log tan 1+log tan 2 + log tan 3 +log 4 + .....+log tan 89 =
1,2,3,4,... , 89 are angles in degrees

Answer 1

Hi,

logtan1+logtan2+logtan3+……………….+logtan89
=log(tan1×tan2×tan3×……………×tan89)

since loga+logb=log(a×b)
we know that tan89=tan(90-1)=cot1
Tan89=cot1
tan88=cot2
tan87=cot3
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We know that tan@×cot@=1

Therefore,
tan1×tan89=1
tan2×tan88=2
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Therefore the given question will be
log(1×1×1×1×1............×1)=log1=0.


Hope you find this helpful.

Answer 2

we know that log a + log b = log ab.

=>log ( tan 1× tan 2× tan 3 × ............tan 89)

=>log ( tan 1 × tan 2×....tan 44 ×cot 44× cot 43....cot 2 × cot 1). {tan a = cot 90-a}

=> log( 1×1×1×1...×1)

=>log (1)

=> 0.