Question

Find the value of $\log \tan 1^{\circ }+\log \tan 2^{\circ }+\log \tan 3^{\circ }+.....+\log \tan {89}^{\circ }$

Answer 1

$\log \tan 1^{\circ }+\log \tan 2^{\circ }+\log \tan 3^{\circ }+.....+\log \tan {89}^{\circ }$
$=\log (\tan 1^{\circ }.\tan 2^{\circ }.\tan 3^{\circ }....\tan {45}^{\circ }....\tan {89}^{\circ })$
$=\log (\tan 1^{\circ }.\tan 2^{\circ }.\tan 3^{\circ }...\tan {45}^{\circ }...\tan 3^{\circ }\cot 2^{\circ }\cot 1^{\circ })$
$=\log \left(\mathrm{1.1.1....1}\log 1\right)$

$=0$