How do I compute this limit?

[opticaltempest]

How can I compute the following limit? I am looking for a trigonometry identity but cannot see any that might work.

\(\displaystyle \L
{ {\lim }\limits_{\Delta t \to 0} \frac{{\sin \left( {t + \Delta t} \right) - \sin t}}{{\Delta t}}}\)


Thanks

 

[pka]

You must recall that \(\displaystyle \L
\lim _{\Delta x \to 0} \frac{{\sin (\Delta x)}}{{\Delta x}} = 1\quad \& \quad \lim _{\Delta x \to 0} \frac{{\cos (\Delta x) - 1}}{{\Delta x}} = 0\)


Then \(\displaystyle \L
\begin{array}{rcl}
\frac{{\sin (t + \Delta x) - \sin (x)}}{{\Delta x}} & = & \frac{{\sin (t)\cos (\Delta x) + \sin (\Delta x)\cos (t) - \sin (t)}}{{\Delta x}} \\
& = & \sin (t)\frac{{\cos (\Delta x) - 1}}{{\Delta x}} - \left( {\frac{{\sin (\Delta x)}}{{\Delta x}}} \right)\cos (t) \\
\end{array}\)

 

[opticaltempest]

Thanks PKA. I disapointed that I didn't see that I needed to factor sin(a)sin(b)-sin(a) to (sin(a))(sin(b)-1).