Suppose \(0^\circ<\theta<90^\circ\) and \(\tan\theta+\cot\theta=\displaystyle\frac{10}{3}\). Find \(\sin\theta\cos\theta\).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{3}{10}\)
Write \(\tan\theta\) and \(\cot\theta\) with sine and cosine and add. The numerator is \(\sin^2\theta+\cos^2\theta=1\), so \(\tan\theta+\cot\theta=\displaystyle\frac{1}{\sin\theta\cos\theta}\).
\(\tan\theta+\cot\theta=\displaystyle\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}=\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}=\frac{1}{\sin\theta\cos\theta}\)
\(\displaystyle\frac{1}{\sin\theta\cos\theta}=\frac{10}{3}\), so \(\sin\theta\cos\theta=\displaystyle\frac{3}{10}\)
Such an angle exists: \(\tan\theta=3\) (or \(\frac{1}{3}\)) gives \(3+\frac{1}{3}=\frac{10}{3}\).