Question : If $\operatorname{cosec} A+\cot A=7$, then $\operatorname{cosec} A$ is equal to:
Option 1: $\frac{11}{7}$
Option 2: $\frac{16}{7}$
Option 3: $\frac{25}{7}$
Option 4: $\frac{19}{7}$
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Correct Answer: $\frac{25}{7}$
Solution :
We know if $\operatorname{cosec}x + \cot x = a$, then $\operatorname{cosec} x - \cot x = \frac{1}{a}$
Given, $\operatorname{cosec} A+\cot A=7$ ........(1)
⇒ $\operatorname{cosec} A-\cot A=\frac17$ .........(2)
Solving these equations to get the value of $\operatorname{cosec} A$
(1) + (2), we get,
⇒ $\operatorname{cosec} A+\cot A+\operatorname{cosec} A-\cot A=7+\frac{1}{7}$
⇒ $2\operatorname{cosec} A = \frac{49+1}{7}$
⇒ $2\operatorname{cosec} A = \frac{50}{7}$
$\therefore \operatorname{cosec} A = \frac{25}{7}$
Hence, the correct answer is $\frac{25}{7}$.
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