$$0.2^{\cos(2x)}-\frac1{25^{\cos^2(x)}}<4\times 125^{-\left(\frac12\right)}$$
I got it down to this:
$$5^{1-2\sin^2(x)}-5^{-2\cos^2(x)}<4\times 5^{-\left(\frac32\right)}$$
I don't know what next.
They all have the same base, but I have a sum on the left side and a coefficient on the right that are preventing me from applying the $\log$ function to both sides.
What do I do here?
The trigonometry isn't the problem here, I just need to remove the exponential part of this in-equation and turn it into what will probably be a quadratic trigonometric inequality.
Conveniently, the $10$ pages of my textbook that probably deal with this have been ripped out. -.-
Hint -
Change $1-2\sin^2x$ into $2\cos^2x-1$. Then try to proceed.