#cos alpha=x/(y+z), cos beta=y/(z+x), cos gamma=z/(x+y),#show that #tan^2 (alpha/2)+tan^2 (beta/2)+tan^2 (gamma/2)=1#?

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Answer 1 · 2017-10-24T15:54:41

We have

#cosalpha=x/(y+z)#

#=>1/cosalpha=(y+z)/x#

By dividendo and componendo we get

#(1-cosalpha)/(1+cosalpha)=(y+z-x)/(x+y+z)#

#=>(2sin^2(alpha/2))/(2cos^2(alpha/2))=(y+z-x)/(x+y+z)#

#=>tan^2(alpha/2)=(y+z-x)/(x+y+z)#

Similarly

#tan^2(beta/2)=(x+z-y)/(x+y+z)#

And

#tan^2(gamma/2)=(x+y-z)/(x+y+z)#

Hence

#tan^2(alpha/2)+tan^2(beta/2)+tan^2(gamma/2)=(x+y+z)/(x+y+z)=1 #