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

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if a,b,c are angles of triangle such that a is obtuse, Then show that TanBTanC <1
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Savannah 3 weeks 2021-08-20T16:01:28+00:00 1 Answer 0 views 0

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  1. Savannah
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    2021-08-20T16:03:12+00:00
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    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 .

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