First slide

Compound angles

Question

If A,B,C are angles of a triangle such that A is obtuse then

Easy

A

$\tan A \tan B > 1$
B

$\tan B \tan C < 1$
C

$\tan C \tan A > 1$
D

$\tan A \tan B \tan C > 1$

Solution

$A i s o b t u s e \Rightarrow B + C < 90 \Rightarrow \tan (B + C) > 0 \Rightarrow \frac{\tan A + \tan B}{1 - \tan A \tan B} > 0 \Rightarrow 1 - \tan A \tan B > 0 (∵ \tan A + \tan B > 0) \Rightarrow \tan A \tan B < 1$

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