Discussion
If \(\sin\theta \cos\theta \gt 0\) , then \(\theta\) lies
Answer: E
\(\theta\) lies in the first quadrant or in the third quadrant.
The product \(\sin\theta\cos\theta\) is positive when the two factors have the same sign. Both are positive in the first quadrant, and both are negative in the third quadrant. Equivalently, \(\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta\), and \(\tan\theta \gt 0\) in exactly these quadrants.
In the second and fourth quadrants the signs differ, so the product is negative, ruling out the options naming them. 'Only in the first quadrant' misses the third.
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