Question : If $\tan\theta+\sec\theta=3$, $\theta$ being acute, the value of $5\sin\theta$ is:
Option 1: $\frac{5}{2}$
Option 2: $\frac{\sqrt{3}}{5}$
Option 3: $\frac{5}{\sqrt{3}}$
Option 4: $4$
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Correct Answer: $4$
Solution : Given: $\theta$ being acute. $⇒\tan\theta+\sec\theta=3$ ... (1) We know the identity, $⇒\sec^2\theta-\tan^2\theta=1$ $⇒(\tan\theta+\sec\theta)( \sec\theta-\tan\theta)=1$ $⇒\sec\theta-\tan\theta=\frac{1}{3}$ ... (2) Solving equation (1) and (2), $⇒2\sec\theta=3+\frac{1}{3}$ $⇒2\sec\theta=\frac{10}{3}$ $\therefore\sec\theta=\frac{5}{3}$ $⇒\cos \theta=\frac{3}{5}$ $⇒ {\sin \theta=\sqrt{1-\cos^2\theta}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}}$ $\therefore 5\sin \theta=5×\frac{4}{5}=4$ Hence, the correct answer is $4$.
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