Question : If $4\left(\operatorname{cosec}^2 57^{\circ}-\tan ^2 33^{\circ}\right)-\cos 90^{\circ}-y \tan ^2 66^{\circ} \tan ^2 24^{\circ}=\frac{y}{2}$, the value of $y$ is:
Option 1: $\frac{8}{3}$
Option 2: $\frac{3}{8}$
Option 3: $8$
Option 4: $\frac{1}{3}$
Correct Answer: $\frac{8}{3}$
Solution :
$\operatorname{cosec}(90^{\circ} - \theta) = \sec \theta$
$\tan\theta = \frac{1}{\cot\theta}$
So, $4\left(\operatorname{cosec}^2 57^{\circ}-\tan ^2 33^{\circ}\right)-\cos 90^{\circ}-y \tan ^2 66^{\circ} \tan ^2 24^{\circ}=\frac{y}{2}$
⇒ $(\operatorname{cosec}^{2}(90-33)^{\circ} - \tan^{2}33^{\circ}) - 0- y×\tan^{2}66^{\circ} × \tan^{2}(90-66)^{\circ}$ = $\frac{y}{2}$
⇒ $4(\sec^{2}57^{\circ}- \tan^{2}33^{\circ}) - y × \tan^{2}66^{\circ} × \cot^{2}66^{\circ} = \frac{y}{2}$
⇒ $4 - y = \frac{y}{2}$
⇒ y = $\frac{8}{3}$
Hence, the correct answer is $\frac{8}{3}$.
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