Evaluating $\lim_{x\to0}\,3^{(1-\sec^220x)/(\sec^210x-1)}$

Question

I want to evaluate $$\lim_{x\to0}\;3^{(1-\sec^220x)/(\sec^210x-1)}$$

So far these have been my ideas, feel free to correct me:

1. Find that if directly applied to the function, $x_0$ will cause indetermination.
2. Manipulate the function trigonometrically, since $\sec x=\frac{1}{\cos x}$, giving me: $1-\frac{1}{\cos^220x}$ and $\frac{1}{\cos^210x}-1$ respectively in the numerator and denominator of the exponent.
3. Apply the property $\lim_{x\rightarrow0}a^{f(x)}=a^{\lim_{x\rightarrow0}f(x)}$
4. Then apply the property that allows me to split the limit of f/g(x) into \lim f(x)/\lim g(x)
5. This is the last step I thought of, finding the LMC.

I'm not sure where to go now, so I'd appreciate some insight.

Answer 1

$$\begin{align*} \frac{1 - \sec^2(20x)}{\sec^2(10x) - 1} &= \frac{\cos^2(20x) - 1}{1 - \cos^2(10x)} \times \frac{\cos^2(10x)}{\cos^2(20x)} \\ &= -\frac{\sin^2(20x)}{\sin^2(10x)} \times \frac{\cos^2(10x)}{\cos^2(20x)} & \sin^2(\theta)+\cos^2(\theta)=1\\ &= -\frac{4\sin^2(10x)\cos^2(10x)}{\sin^2(10x)} \times \frac{\cos^2(10x)}{\cos^2(20x)} & \sin(2\theta)=2\sin(\theta)\cos(\theta)\\ &=-\frac{4\cos^4(10x)}{\cos^2(20x)} \end{align*}$$