Derivative of Cosine Function/Proof 3

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Theorem

$\map {\dfrac \d {\d x} } {\cos x} = -\sin x$


Proof

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

$\blacksquare$