Baffling identity: \(\displaystyle\prod{{\log}_{{{10}}}{\tan{=}}}\sum{{\log}_{{{10}}}{\tan{}}}\) I am quite a baffled

DeamyKetSate1m

DeamyKetSate1m

Answered question

2022-03-19

Baffling identity: log10tan=log10tan 
I am quite a baffled now, I am not getting by how it can be written that :
log10tan40°·log10tan41°·log10tan42°·log10tan43°log10tan50°¯log10tan40°

+log10tan41°+log10tan42°+log10tan43°++log10tan50°
Is it even valid ? If yes,how ?

Answer & Explanation

Lindsey Rocha

Lindsey Rocha

Beginner2022-03-20Added 4 answers

Yes it is valid.
log10tan40+log10tan50=log10tan40+log10cot40=log101=0
Similarly combine 41 and 49, 42 and 48 etc.
The product on the left side is 0 as log10tan45=0
RI5N6mv3

RI5N6mv3

Beginner2022-03-21Added 4 answers

HINT: Observe that
log10tan{40}+log10tan{50}=log10sin{40}log10cos{40}+log10sin{50}log10cos{50}
Now use the fact that sin(90x)=cosx and see that they cancel out.

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