a: \(\dfrac{sina+sin2a}{1+cosa+cos2a}=\dfrac{sina+2\cdot sina\cdot cosa}{1+cosa+2\cdot cos^2a-1}\)
\(=\dfrac{sina\left(2\cdot cosa+1\right)}{2\cdot cos^2a+cosa}=\dfrac{sina\left(2\cdot cosa+1\right)}{cosa\left(2\cdot cosa+1\right)}=\dfrac{sina}{cosa}=tana\)
b: \(\dfrac{cos\left(\dfrac{\Omega}{4}+a\right)-cos\left(\dfrac{\Omega}{4}-a\right)}{sin\left(\dfrac{\Omega}{4}+a\right)-sin\left(\dfrac{\Omega}{4}-a\right)}\)
\(=\dfrac{cos\left(\dfrac{\Omega}{4}\right)\cdot cosa-sin\left(\dfrac{\Omega}{4}\right)\cdot sina-cos\left(\dfrac{\Omega}{4}\right)\cdot cosa-sin\left(\dfrac{\Pi}{4}\right)\cdot sina}{sin\left(\dfrac{\Pi}{4}\right)\cdot cosa+sina\cdot cos\left(\dfrac{\Pi}{4}\right)-sin\left(\dfrac{\Pi}{4}\right)\cdot cosa+sina\cdot cos\left(\dfrac{\Pi}{4}\right)}\)
\(=\dfrac{-2\cdot sina\cdot\dfrac{\sqrt{2}}{2}}{2\cdot sina\cdot\dfrac{\sqrt{2}}{2}}=-1\)
c: \(\dfrac{sin^2a}{4-4\cdot sin^2\left(\dfrac{a}{2}\right)}=\dfrac{sin^2a}{4\left(1-sin^2\left(\dfrac{a}{2}\right)\right)}=\dfrac{sin^2a}{4\cdot cos^2\left(\dfrac{a}{2}\right)}\)
\(=\dfrac{\left(sin\left(2\cdot\dfrac{a}{2}\right)\right)^2}{4\cdot cos^2\left(\dfrac{a}{2}\right)}=\dfrac{4\cdot sin^2\left(\dfrac{a}{2}\right)\cdot cos^2\left(\dfrac{a}{2}\right)}{4\cdot cos^2\left(\dfrac{a}{2}\right)}=sin^2\left(\dfrac{a}{2}\right)\)
