Originally Posted by
HallsofIvy Wouldn't it be better for you to proe them? Do you know identities for trig functions of multiple or fractional arguments?
$\displaystyle sin(2x)= 2sin(x)cos(x)$ and $\displaystyle cos(2x)= cos^2(x)- sin^2(x)$
In the second, we can replace $\displaystyle sin^2(x)$ by $\displaystyle 1- cos^2(x)$ to get $\displaystyle cos(2x)= cos^2(x)- (1- cos^2(x))= 2cos^2(x)- 1$
so that $\displaystyle cos^2(x)= (1+ cos(2x))/2$, $\displaystyle cos(x)= \pm \sqrt{(1/2)(1+ cos(2x))}$ and letting $\displaystyle \theta= 2x$ that becomes $\displaystyle cos(\theta/2)= \pm\sqrt{(1/2)(1+ cos(2\theta))}$. Similarly, $\displaystyle sin(2x)= 2sin(x)cos(x)$ and $\displaystyle sin(\theta/2)= \pm\sqrt{(1/2)(1- cos(\theta))}$.
Those, together with the fact that $\displaystyle tan(x)= \frac{sin(x)}{cos(x)}$, are what you need.