If \(A + B + C = \pi\) \((A, B, C > 0)\) and the angle C is obtuse, then which of the following is/are correct? I. \(\sin A . \sin B < 1\) II. \(\tan A . \tan B > 1\) Select the answer using the code given below:
I only
Since C is obtuse, \(C > \frac{\pi}{2}\), so \(A + B = \pi - C < \frac{\pi}{2}\), with \(A, B > 0\). As both A and B are then positive acute angles with \(A + B < \frac{\pi}{2}\), each of \(\sin A, \sin B < 1\), so \(\sin A . \sin B < 1\) (Statement I true). Also, since \(A + B < \frac{\pi}{2}\), \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A . \tan B} > 0\); since the numerator is positive, the denominator must also be positive, giving \(\tan A . \tan B < 1\) (Statement II false). So only I is correct.
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