Question

if a,b,c are angles of triangle such that a is obtuse, Then show that TanBTanC <1
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Answer 1

It has given that a, b, c are angles of a triangle such that a is obtuse angle.

To show that : Tanb . tanc < 1

solution : as it has given that a is obtuse angle

so, π/2 < a < π

⇒π/2 < π - (b + c) < π. [ because a + b + c = π ]

⇒π/2 - π < -(b + c) < π - π

⇒-π/2 < -(b + c) < 0

⇒0 < (b + c) < π/2

⇒(b + c) < π/2

⇒b < π/2 - c

⇒tanb < tan(π/2 - c)

⇒tanb < cotc

⇒tanb < 1/tanc [ assuming value of tanc > 1]

⇒tanb tanc < 1

hence proved .