Question : If $\sec A+\tan A=5$, then $\sin A$ is equal to:
Option 1: $\frac{5}{13}$
Option 2: $\frac{5}{12}$
Option 3: $\frac{13}{12}$
Option 4: $\frac{12}{13}$
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Correct Answer: $\frac{12}{13}$
Solution :
Given: $\sec A+\tan A=5$ ----------(equation 1)
We know,
$\sec^2A–\tan^2A=1$
⇒ $(\sec A+\tan A)(\sec A–\tan A)=1$
⇒ $5(\sec A–\tan A)=1$
⇒ $\sec A–\tan A=\frac{1}{5}$ -----------(equation 2)
Solving equations 1 and 2 we get,
$\sec A=\frac{26}{10}$ and $\tan A=\frac{24}{10}$.
So, $\sin A=\frac{\tan A}{\sec A}=\frac{\frac{24}{10}}{\frac{26}{10}}=\frac{12}{13}$.
Hence, the correct answer is $\frac{12}{13}$.
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