Question

If x, y, z be all positive acute angles, then find the least value of tan x (cot y + cot z) + tan y (cot z + cotx) +
tan z (cot x + cot y).
y)​

Answer 1

Answer:

If a number m is positive so

m+

m

1

≥2

So, we will use this rule here,

tanx(coty+cotz)+tany(cotz+cotx)+tanz(cotx+coty)

=tanx(

tany

1

+

tanz

1

)+tany(

tanz

1

+

tanx

1

)+tanz(

tanx

1

+

tany

1

)

=

tany

tanx

+

tanz

tanx

+

tanz

tany

+

tanx

tany

+

tanx

tanz

+

tany

tanz

=(

tany

tanx

+

tanx

tany

)+(

tanz

tanx

+

tanx

tanz

)+(

tanz

tany

+

tany

tanz

)

=(a+

a

1

)+(b+

b

1

)+(c+

c

1

)

where a=

tany

tanx

,b=

tanz

tany

,c=

tanx

tanz

≥2+2+2(from above)

≥6

So, minimum value is 6