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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