Question : If $\operatorname{cosec} \theta=1 \frac{7}{22}$, then find the value of $\cot ^2 \theta$.
Option 1: $\frac{49}{484}$
Option 2: $\frac{357}{484}$
Option 3: $\frac{225}{484}$
Option 4: $\frac{7}{22}$
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Correct Answer: $\frac{357}{484}$
Solution :
Given: $\operatorname{cosec} \theta=1 \frac{7}{22}$
⇒ $\operatorname{cosec} \theta=\frac{29}{22}$
We know that: $\operatorname{cosec}^{2}\theta=\cot^{2}\theta+1$
⇒ $\cot^{2}\theta=\frac{841}{484}–1$
⇒ $\cot^{2}\theta=\frac{841–484}{484}$
⇒ $\cot^{2}\theta=\frac{357}{484}$
Hence, the correct answer is $\frac{357}{484}$.
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