- #1

MathematicalPhysicist

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1)let a,b,c be acute angles, if tg(a)tg(b)tg(c)=1 then sin(a)sin(b)sin(c)<=1/2sqrt2

2) prove that for every x,y cos(x^2)+cos(y^2)-cos(xy)<3

for the second question i tried to use the fact that (x^2+y^2)/2>=xy and the fact that on some intervals the function cos is decreasing, actually what i need to prove is that -cos(xy)<1, cause cosx^2+cosy^2<=2, so i also tried to show that cos(xy) cannot be equal -1, but didnt get much with that.

for the first i used the cosine law and sine law, but also didnt get far, i need to show that (sin^2a(sin^2b+sin^2c)+sin^2b(sin^2a+sin^2c)+sin^2c(sin^2a+sin^2b)<=sin(a)sin(b)sin(c), but i didnt succeded in it.

p.s

if someone wonders why im posting questions of these type is because im reviewing a little bit inequalities.