1. ## Measure-angles

a,b,c measure angles of triangle.
Prove :
$\displaystyle \tan \left( {\frac{a}{2}} \right) + \tan \left( {\frac{b}{2}} \right) + \tan \left( {\frac{c}{2}} \right) = 4\frac{{1 + \sin \left( {\frac{a}{2}} \right)\sin \left( {\frac{b}{2}} \right)\sin \left( {\frac{c}{2}} \right)}}{{\sin a + \sin b + \sin c}}$

2. $\displaystyle \tan\frac{a}{2}+\tan\frac{a}{2}+\tan\frac{c}{2}=\f rac{\sin\frac{a}{2}}{\cos\frac{a}{2}}+\frac{\sin\f rac{b}{2}}{\cos\frac{b}{2}}+\frac{\sin\frac{c}{2}} {\cos\frac{c}{2}}=$

$\displaystyle =\frac{\sin\frac{a+b}{2}}{\cos\frac{a}{2}\cos\frac {b}{2}}+\frac{\sin\frac{c}{2}}{\cos\frac{c}{2}}=$

$\displaystyle =\frac{\cos\frac{c}{2}}{\cos\frac{a}{2}\cos\frac{b }{2}}+\frac{\sin\frac{c}{2}}{\cos\frac{c}{2}}=$

$\displaystyle =\frac{\cos^2\frac{c}{2}+\sin\frac{c}{2}\cos\frac{ a}{2}\cos\frac{b}{2}}{\cos\frac{a}{2}\cos\frac{b}{ 2}\cos\frac{c}{2}}=$

$\displaystyle =\frac{1-\sin^2\frac{c}{2}+\sin\frac{c}{2}\cos\frac{a}{2}\c os\frac{b}{2}}{\cos\frac{a}{2}\cos\frac{b}{2}\cos\ frac{c}{2}}=$

$\displaystyle =\frac{1+\sin\frac{c}{2}\left(\cos\frac{a}{2}\cos\ frac{b}{2}-\cos\frac{a+b}{2}\right)}{\cos\frac{a}{2}\cos\frac {b}{2}\cos\frac{c}{2}}=$

$\displaystyle =\frac{1+\sin\frac{a}{2}\sin\frac{b}{2}\sin\frac{c }{2}}{\cos\frac{a}{2}\cos\frac{b}{2}\cos\frac{c}{2 }}=$

But it is well known that $\displaystyle \sin a+\sin b+\sin c=4\cos\frac{a}{2}\cos\frac{b}{2}\cos\frac{c}{2}=$

3. reddog, it seems too easy for you . . . . well done