$\log \tan 3 ^\circ+\log \tan 6 ^\circ+ \cdots +\log \tan 87 ^\circ$

tan(3)=cot(87); tan(87)=*cot(3) (co-functions=> "tan(x)=cot(90-x)");

thus log tan3+log tan 87 = log tan(3)tan(87) = log (tan3)(*cot3) = log1= 0, log tan(6)+log tan(84) = 0 until you reach the two middlemost terms... by applying it to all pairs you'll get 0+0+0+0....+0

Thus the sum is 0.

If log tanx=a, then log tan(90-x)=(-a).

Thus log tan3+log tan 87=0, log tan6 + log tan84=0 and son on till log tan42 +log tan 48 = 0. and log tan45 = log 1= 0.

Hence required answer = 0

\[\begin{align} &= \log {\tan{3^{\circ}}} + \log {\tan{6^{\circ}}} + \ldots + \log {\tan{45^{\circ}}} + \ldots + \log {\tan{84^{\circ}}} + \log {\tan{87^{\circ}}} \\ \\ &= \log {\bigg(\tan{3^{\circ}} \cdot \tan{6^{\circ}} \cdots \tan{45^{\circ}} \cdots \tan{84^{\circ}} + \tan{87^{\circ}}\bigg)} \quad \quad \color{Blue}{[\log {(ab)} = \log {(a)} + \log {(a)} ]} \\ \\ &= \log {\bigg(\tan{3^{\circ}} \cdot \tan{6^{\circ}} \cdots \tan{45^{\circ}} \cdots \cot{6^{\circ}} + \cot{3^{\circ}}\bigg)} \\ \\ &= \log {\bigg(\big(\tan{3^{\circ}} \cdot \cot{3^{\circ}} \big) \cdot \big(\tan{6^{\circ}} \cdot \cot{6^{\circ}} \big) \cdots \big(\tan{45^{\circ}} \big) \bigg)} \\ \\ &= \log {\bigg( \cancel{\big(\tan{3^{\circ}} \cdot \cot{3^{\circ}} \big)} \cdot \cancel{\big(\tan{6^{\circ}} \cdot \cot{6^{\circ}} \big)} \cdots \big(\tan{45^{\circ}} \big) \bigg)} \quad \quad \color{Blue}{[\tan{c^{\circ}} \cdot \cot{c^{\circ}} \implies \cancel{\tan{c^{\circ}}} \cdot \frac{1}{\cancel{\tan{c^{\circ}}}}= 1]} \\ \\ &= \log {(\tan{45^{\circ}})} \\ \\ &= \log {(1)} \\ \\ &= \boxed{0} \quad \quad \color{Blue}{[\log_{d} {1} = 0]}

\end{align}\]

Log tan3+log tan6...log tan87 =Log(tan3 tan6...tan87)

We know that tan(87)=cot(3)

So, tan(87)tan(3)=1 That's same for all other pairs

So, this gives us Log(1 * 1 * 1...* 1)=log(1)=0

The given expression can be simplified to $\log\left(\tan 3 × \tan 6 × \cdots × \tan87\right) = \log 1 = \color{#69047E}{\boxed{0}}$ .

sum of the terms =(first term+last term) over 2 log tan 3 + log tan 87=log tan 90 log tan 90 /2 =log tan 45 =log 1 (since tan 45 =1) =0