Derivative of Cosine Function/Proof 3

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Theorem

$D_x \left({\cos x}\right) = -\sin x$


Proof

\(\displaystyle D_x \cos x\) \(=\) \(\displaystyle D_x \, \map \sin {\frac \pi 2 - x}\) Sine of Complement equals Cosine
\(\displaystyle \) \(=\) \(\displaystyle -\map \cos {\frac \pi 2 - x}\) Derivative of Sine Function and Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle -\sin x\) Cosine of Complement equals Sine

$\blacksquare$