Question : If $\tan \theta+\sec \theta=7, \theta$ being acute, then the value of $5 \sin \theta$ is:
Option 1: $\frac{25}{24}$
Option 2: $\frac{24}{25}$
Option 3: $\frac{1}{24}$
Option 4: $\frac{24}{5}$
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Correct Answer: $\frac{24}{5}$
Solution :
Given: $\tan\theta+\sec\theta=7$ --------------------(1)
We know that $\sec^2\theta-\tan^2\theta=1$
⇒ $(\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1$
⇒ $7×(\sec\theta-\tan\theta)=1$
⇒ $\sec\theta-\tan\theta=\frac{1}{7}$ --------------------(2)
Adding equations (1) and (2) we get,
$2\sec\theta=7+\frac{1}{7}$
⇒ $\sec\theta=\frac{25}{7}$
⇒ $\cos\theta=\frac{7}{25}$
⇒ $\sin\theta=\sqrt{1-(\frac{7}{25})^2}$
⇒ $\sin\theta=\frac{24}{25}$
⇒ $5\sin\theta=\frac{24}{5}$
Hence, the correct answer is $\frac{24}{5}$.
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