Question : If $\tan (11 \theta)=\cot (7 \theta)$, then what is the value of $\sin ^2(6 \theta)+\sec ^2(9 \theta)+\operatorname{cosec}^2(12 \theta) ?$
Option 1: $\frac{23}{6}$
Option 2: $\frac{35}{12}$
Option 3: $\frac{31}{12}$
Option 4: $\frac{43}{12}$
Correct Answer: $\frac{43}{12}$
Solution : $\tan (11 \theta)=\cot (7 \theta)$ $⇒\tan (11 \theta)=\tan (90^\circ-7 \theta)$ $⇒11 \theta=90^\circ-7 \theta$ $⇒18 \theta=90^\circ$ $⇒ \theta=5^\circ$ $\sin ^2(6 \theta)+\sec ^2(9 \theta)+\operatorname{cosec}^2(12 \theta)$ Substituting $ \theta=5^\circ$, $=\sin ^2(30 ^\circ)+\sec ^2(45 ^\circ)+\operatorname{cosec}^2(60 ^\circ)$ $=(\frac{1}{2})^2+(\sqrt2)^2+(\frac{2}{\sqrt3})^2$ $=\frac{1}{4}+2+\frac{4}{3}$ $=\frac{43}{12}$ Hence, the correct answer is $\frac{43}{12}$.
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