Question

Find the value of $\tan 1^0\tan 2^0\tan 3^0.......\tan {89}^0$.

Answer 1

$\tan {89}^0=\tan ({90}^0-1^0)=\cot 1^0\tan {88}^0=\tan ({90}^0-2^0)=\cot 2^0$

Now

$\tan 1^0\tan 2^0\tan 3^0.....\tan {44}^0\tan {45}^0\tan {46}^0...........\tan {87}^0\tan {88}^0\tan {89}^0=\tan 1^0\tan 2^0\tan 3^0.....\tan {44}^0\times 1\times \cot {44}^0.......\cot 3^0\cot 2^0\times \cot 1^0=1\times 1\times \mathrm{1.....}\times 1=1$