Cotangent of 45 Degrees

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Theorem

$\cot 45^\circ = \cot \dfrac {\pi} 4 = 1$

where $\cot$ denotes cotangent.


Proof

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \cot 45^\circ\) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \frac {\cos 45^\circ} {\sin 45^\circ}\) \(\displaystyle \) \(\displaystyle \)          Cotangent is Cosine divided by Sine          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle \frac{\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}\) \(\displaystyle \) \(\displaystyle \)          Cosine of 45 Degrees and Sine of 45 Degrees          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\) \(\displaystyle \) \(\displaystyle 1\) \(\displaystyle \) \(\displaystyle \)          multiplying top and bottom by $\dfrac {\sqrt 2} 2$          

$\blacksquare$


Sources