Can we solve $2 \int_0^{\frac{\pi}{4}} \sin^2x dx$ using any property also? $$\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \sin^2 x dx$$
My way of solving - 
$$2 \int_0^{\frac{\pi}{4}} \frac{1- \cos2x}{2} dx$$
On solving i get $\dfrac{\pi}4 - \dfrac 12$ as an answer.

My main question is can we us any other property here also?

Something like -
$$2 \int_0^{\frac{\pi}{4}} \sin^2x dx$$
$$2 \int_0^{\frac{\pi}{4}} \sin^2(\frac{\pi}{4}-x)dx$$
Then how to proceed?

The method you suggest is the best, but if you really want a different method, you could try integration by parts to get $$I=-\sin x\cos x+\int \cos^2 x dx$$ and then use $\cos^2 x =1-\sin^2 x$ to get an expression for $2I$ And so on...
