Apparently one option is to differentiate the identity $ \sin(2x) \equiv 2 \sin(x)\cos(x) $ to get the identity $\cos(2x) \equiv \cos^2x – \sin^2x $. Which is surprising as I didn't realize that differentiating an identity produces another identity.
However, I'd like to know how it can be done without involving calculus.
I have the list of all trigonometric properties and operations at hand, but can't find the right way to relate them to go from one to the other.
Could I get a hint?
Best Answer
$$\cos^2 (2x) = 1 - \sin^2 (2x) = 1 - 4 \sin^2 x \cos^2 x = 1 - 4 \sin^2 x (1 - \sin^2 x)$$ $$ = 4 \sin^4 x - 4 \sin^2 x +1$$
Now if we let $u = \sin x$, we have $4u^4 - 4u^2 + 1 = (2u^2-1)^2$. Thus:
$$\cos^2 (2x) = (2 \sin^2 x - 1)^2$$ $$\cos^2 (2x) = (2 \sin^2 x - (\sin^2 x +\cos^2 x) )^2$$
and now you are very close to the desired result.
One last note: from $a^2 = b^2$ we cannot directly conclude that $a = b$. However, we can disprove that $\cos(2x) = (\sin^2 x - \cos^2 x)$ by substituting in $x = 0$ for example: this gives $1 = -1$ which is false. Since what we have done is true no matter which $x$ we choose, the other possibility $\cos(2x) = -(\sin^2 x - \cos^2 x) = \cos^2 x - \sin^2 x$ must be true.