Question

Find the exact value of tan 195°.

Answer 1

`\tan 195^(\circ)\\\\=\tan\left(3 * 90^(\circ) - 75^(\circ) \right)\\\\=\cot 75^(\circ)\\\\=\left(\tan 75^(\circ) \right)^(-1)\\\\= \textbf{[}\tan \left(30^(\circ) + 45^(\circ)\right) \right \textbf{]}^(-1)\\\\=\left((\tan 30^(\circ) +\tan 45^(\circ))/(1- \tan 45^(\circ) \cdot \tan 30^(\circ)) \right)^(-1)\\\\\\=\left( \frac{\frac 1{\sqrt 3}+1}{1- \frac 1{\sqrt 3}} \right)^(-1)\\\\\\=\left( (\sqrt 3+1)/(\sqrt 3-1)\right)^(-1)\\\\\\=(\sqrt 3-1)/(\sqrt 3 +1)\\`

`=(\left( \sqrt 3-1\right)^2)/(\left( \sqrt 3+1\right)\left( \sqrt 3-1\right))\\\\\\=(3+1-2\sqrt 3)/(3-1)\\\\\\=(4-2\sqrt 3)/(2)\\\\\\=2-\sqrt 3`