Tangent of 45 Degrees

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Theorem

$\tan 45 \degrees = \tan \dfrac \pi 4 = 1$

where $\tan$ denotes tangent.


Proof

\(\ds \tan 45 \degrees\) \(=\) \(\ds \frac {\sin 45 \degrees} {\cos 45 \degrees}\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {\frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}\) Sine of $45 \degrees$ and Cosine of $45 \degrees$
\(\ds \) \(=\) \(\ds 1\) dividing top and bottom by $\sqrt 2 / 2$

$\blacksquare$


Sources