Why should we have $\sin^2(x) = \frac{1-\cos(2x)}{2}$ knowing that $\sin^2(x) = 1 - \cos^2(x)$? 
Why should we have $\sin^2(x) = \frac{1-\cos(2x)}{2}$ knowing that $\sin^2(x) = 1 - \cos^2(x)$?

Logically, can you not subtract $\cos^2(x)$ to the other side from this Pythagorean identity $\sin^2(x)+\cos^2(x)=1?$ 
When I look up trig identities, however, it says $\sin^2(x) = \frac{1-\cos(2x)}{2}$. 
Why is this?

Both formulas are true, however, both are useful in different contexts (applications).


*

*You use $\sin^2(x) = \frac{1-\cos(2x)}{2}$ for integrating $\sin^2(x)$. 

*You use $\sin^2(x) = 1 - \cos^2(x)$, for example, when solving $\sin^2(x) = 2\cos(x)$. 
Note that it is just in some way more "natural" to write $\sin^2(x) + \cos^2(x)=1$, because this gives both $\sin^2(x) = 1 - \cos^2(x)$ and $\cos^2(x) = 1 - \sin^2(x)$ in one "natural looking" formula.
