Tangent of 225 Degrees

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Theorem

$\tan 225^\circ = \tan \dfrac {5 \pi} 4 = 1$

where $\tan$ denotes tangent.


Proof

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \tan 225^\circ\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \tan \left({360^\circ - 135^\circ}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle -\tan 135^\circ\) \(\displaystyle \) \(\displaystyle \)          Tangent of Conjugate Angle          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle 1\) \(\displaystyle \) \(\displaystyle \)          Tangent of 135 Degrees          

$\blacksquare$


Sources