Question : If $4\left(\operatorname{cosec}^2 57^\circ-\tan ^2 33^\circ\right)-\cos 90^\circ+y \tan ^2 66^\circ \tan ^2 24^\circ=\frac{y}{2}$, then the value of $y$ is:
Option 1: 4
Option 2: – 4
Option 3: 8
Option 4: – 8
Correct Answer: – 8
Solution :
$4\left(\operatorname{cosec}^2 57^\circ-\tan ^2 33^\circ\right)-\cos 90^\circ+y \tan ^2 66^\circ \tan ^2 24^\circ=\frac{y}{2}$
Using, $\cos 90^\circ = 0$, $ \operatorname {cosec}^2 A - 1 = \cot^2 A$ and $\tan (90^\circ - A) = \cot A$ where $A$ is an angle.
Substituting these identities into the given equation,
$⇒4\left[1 + \cot^2 57^\circ - (\tan^2 (90^\circ-57^\circ))\right] + y \tan^2 66^\circ \tan^2 (90^\circ - 66^\circ) = \frac{y}{2}$
We know that $\cot A = \tan (90^\circ - A)$
$⇒4\left[1 +\cot^2 57^\circ - \cot^2 57^\circ\right] + y \tan^2 66^\circ \cot^2 66^\circ = \frac{y}{2}$
$⇒4\left[1\right] + y = \frac{y}{2}$
$⇒\frac{y}{2}=-4$
$⇒y = -8$
Hence, the correct answer is '– 8'.
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