First slide

Compound angles

Question

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

Moderate

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

$\begin{array}{l} A > π / 2 B + C < π / 2 \Rightarrow \tan (B + C) > 0 \\ \Rightarrow \frac{\tan B + \tan C}{1 - \tan B \tan C} > 0 \\ \Rightarrow \tan B \tan C < 1 (as \tan B, \tan C > 0) \end{array}$

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