Cosecant of 45 Degrees

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Theorem

$\csc 45^\circ = \csc \dfrac \pi 4 = \sqrt 2$

where $\csc$ denotes cosecant.


Proof

\(\displaystyle \csc 45^\circ\) \(=\) \(\displaystyle \frac 1 {\sin 45^\circ}\) Cosecant is Reciprocal of Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\frac {\sqrt 2} 2}\) Sine of 45 Degrees
\(\displaystyle \) \(=\) \(\displaystyle \sqrt 2\) multiplying top and bottom by $2 \sqrt 2$

$\blacksquare$


Sources