Triangle type from tangent product of two angles

[MathExpert]

Prove that the triangle ABC is:

a) acute, if and only if tan(alfa)*tan(beta)>1
b) right, if and only if tan(alfa)*tan(beta)=1
c) obtuse, if and only if tan(alfa)*tan(beta)<1

 

[tongos]

start with the following...
k=alpha+beta
90<k<180 in an acute triangle
k=90 in a right triangle
k<90 in an obtuse triangle

(sinx/cosx)(sin(k-x)/cos(k-x))
=(sinxsinkcosx-cosksin^2x)/(cos^2xcosk+sinksinxcosx)
sinxsinkcosx=A and cosk=B
(A-Bsin^2x)/(A+Bcos^2x)= (A-Bsin^2x)/(A+B-Bsin^2x)
we then want B equal to zero. cosk=0 when k=90. when k is greater B becomes negative.