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# Proving the derivatives of sin(x) and cos(x)

​Proving that the derivative of sin(x) is cos(x) and that the derivative of cos(x) is -sin(x).
The trigonometric functions sine, left parenthesis, x, right parenthesis and cosine, left parenthesis, x, right parenthesis play a significant role in calculus. These are their derivatives:
\begin{aligned} \dfrac{d}{dx}[\sin(x)]&=\cos(x) \\\\ \dfrac{d}{dx}[\cos(x)]&=-\sin(x) \end{aligned}
The AP Calculus course doesn't require knowing the proofs of these derivatives, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

## First, we would like to find two tricky limits that are used in our proof.

### 1. $\displaystyle\lim_{x\to 0}\dfrac{\sin(x)}{x}=1$limit, start subscript, x, \to, 0, end subscript, start fraction, sine, left parenthesis, x, right parenthesis, divided by, x, end fraction, equals, 1

Limit of sin(x)/x as x approaches 0See video transcript