Limit $\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta }{ \theta } } $ $$\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta  }{ \theta  }  } $$
Steps I took:
$$\lim _{ \theta \rightarrow 0 }{ \frac { 1-2sin^{ 2 }\theta -cos\theta  }{ \theta  }  } =$$
$$\lim _{ \theta \rightarrow 0 }{ \frac { -2sin^{ 2 }\theta  }{ \theta  }  } +\lim _{ \theta \rightarrow 0 }{ \frac { 1-cos\theta  }{ \theta  }  } $$
$$\lim _{ \theta \rightarrow 0 }{ \frac { 1-cos\theta  }{ \theta  } =0 } $$
$$\lim _{ \theta \rightarrow 0 }{ \frac { -2(sin\theta )(sin\theta ) }{ \theta  } = } $$
$$\lim _{ \theta \rightarrow 0 }{ { \quad -2(sin\theta ) }\cdot 1=0 } $$
$$0+0=0$$
Something seems off about the way I went about this but I can't figure it out.

L'Hospital is very efficient here: just by inspection, $\dfrac{-0+0}1$.
Also, as the numerator is an even smooth function, it must be quadratic in $\theta$, hence $0$.
