Logarithms of Tangents

First, we gather all the terms into a single logarithm: $\log(\tan1°)+\log(\tan2°)+\dots+\log(\tan89°)=\log(\tan1°\times\tan2°\times\dots\times\tan89°)$

Then we observe that, because of symmetry of the sine and cosine functions, $\tan1°=\frac{1}{\tan89°}, \tan2°=\frac{1}{\tan88°}$ and so on.

In this way, we have $\tan1°\times\tan89°=\tan2°\times\tan88°=\tan45°=1$ .

Then, we get that: $\log(\tan1°\times\tan2°\times\dots\times\tan89°) \\=\log(1\times1\times\dots\times1) \\=\log(1) \\=0$

We know that $\log (m) + \log (n)$ = $\log (mn)$ and, $\tan \theta = \frac {1}{\cot \theta}$ .

Using the concepts above, it is easy to see that $\log (\tan (1)) + ... \log (\tan (89))$ reduces to $\tan 45$ .

The value of $\tan 45$ is $1$ and we know that $\log (1)$ to the base anything is always $0$ .

Hence, the answer is $0.$

log tan 1 + log tan 2 + log tan 3 + ....................+ log tan 89 = log ( tan 1 x tan 2 x tan 3 x .............. x tan 89) = log { (tan 1 x tan 89) x (tan 2 x tan 88) x ( tan 3 x tan 87) x ................ x (tan 44 x tan 46) x tan 45} = log { ( tan 1 x tan (90 - 1) ) x ( tan 2 x tan ( 90 -2)) x ( tan 3 x tan (90 - 3)) x ............x ( tan 44 x tan (90 - 44)) x tan 45 } = log { (tan 1 x cot 1) x ( tan 2 x cot 2 ) x ( tan 3 x cot 3) x ......x( tan 44 x cot 44 ) x tan 45 } = log { 1 x 1 x 1 x .....................x 1 x tan 45} = log { tan 45} = log 1 = 0

We know $tan(x) = sin(x)/cos(x)$

so:

$log(tan(x)) = log(sin(x)) - log(cos(x))$

We also know that $cos(x) = sin(90^{\circ} - x)$ . This means:

$log(tan(1^{\circ})) + ... + log(tan(89^{\circ}))\\ = log(sin(1^{\circ})) + ... + log(sin(89^{\circ})) - log(cos(1^{\circ})) - ... - log(cos(89^{\circ})\\ = log(sin(1^{\circ})) + ... + log(sin(89^{\circ})) - log(sin(89^{\circ})) - ... - log(sin(1^{\circ}))\\ = 0$

log(tan1)+log(tan2)+tan(tan3)+......+log(tan89)=log(tan1.tan2.tan3....tan89)

=log1

=0

Applied Excel for both addition and multiplication to confirm.

Log 1 = 0

In fact, (Tan x)(Cot x) = 1 for (Pi/ 2 - x) swap between Sin x and Cos x within the range.

$\displaystyle \sum_{n=1}^{89} \log{\tan{n°}}=\log{\prod_{n=1}^{89} \tan{n°}}$

$\tan{1°}\times\tan{89°}=\tan{1°}\cot{1°}=1$

Similarly every term cancels out except $\tan{45°}=1$

So, finally: $\displaystyle \sum_{n=1}^{89} \log{\tan{n°}}=\log{\prod_{n=1}^{89} \tan{n°}}=\log{1}=0$

log(tan1)+log(tan2)+log(tan3)+...+log(tan88)+log(tan89)=(log(tan1)+log(tan89))+(log(tan2)+log(tan88))+...+(log(tan49)+log(tan41)) we have log(a)+log(b)=log(a b)=> log(tan1)+log(tan89)=log(tan1 tan89), we also have tan(1) tan(89)=sin(1) sin(89)/cos(1) cos(89) =(cos(88)-cos(90))/(cos(90)+cos(88))=1 => log(tan1)+log(tan89)=log(tan1 tan89)=log(1)=0 so the total is 0

tan(1) tan(89) = 1, tan(2) tan(88) = 1, ...

we can simplify to log(1) = 0

The sum: $\log(\tan 1^\circ ) + \log (\tan 2^\circ) + \log (\tan 3^\circ) + \ldots + \log (\tan 89^\circ )$ could be expressed as $\log \bigl((\tan 1^\circ) \mathsf{x} (\tan 2^\circ) \mathsf{x} (\tan 3^\circ) \mathsf{x} \ldots \mathsf{x} (\tan 89^\circ)\bigr)$ Which is equal to $\log\biggl(\displaystyle{\prod\limits_{x=1}^{89}\tan x}\biggr)$ We can derive that $\tan 1^\circ \mathsf{x} \tan 89^\circ = \tan 2^\circ \mathsf{x} \tan 88^\circ = \tan 45^\circ = 1$ etc $.$ So $\biggl(\displaystyle{\prod\limits_{x=1}^{89}\tan x}\biggr) = 1$ $\therefore \log\biggl(\displaystyle{\prod\limits_{x=1}^{89}\tan x}\biggr) = \log 1 = 0$

note that tan1.tan2.tan3.............tan89 is 1 since tan1=tan89=cot1 & tanxcotx=1 so tan1.tan2.tan3...........tan89 = tan1.tan89.tan.tan2..........tan44.tan46.tan45=1.1.1.1......tan45 = tan45=1 so log(tan1.tan2.tan3..........tan45)=log(1)=0 (since 10^0=1) so answer is 0

Firstly, let a=log(tan1°)+log(tan2°)+log(tan3°)+..+log(tan89°).

Then from log(x)+log(y)= log(xy), we obtain that

a=log[(tan1°)(tan2°)(tan3°)...(tan89°)].

Also from tan(x-y)=(tanx-tany)/(tanx-tany) and tan(90°)=∞, we can verify that

tan(90°-x)=1/tan(x). Thus, that we get a= log[(tan1°)(tan2°)...(tan44°)(tan45°)tan(90°-46°)...tan(90°-1°)] a=log[(tan1°)(tan2°)(tan3°)...(tan44°)(1)(1/tan44°)...(1/tan3°)(1/tan2°)(1/tan1°)] a=log(1) ∴a=0.

log( tan 45) = 0