Question : The value of the following is:
$\frac{\left (\tan20^{\circ} \right )^{2}}{\left (\operatorname{cosec70^{\circ}} \right )^{2}}+\frac{\left (\cot20^{\circ} \right )^{2}}{\left (\sec70^{\circ} \right )^{2}}+2\tan15^{\circ} \tan45^{\circ} \tan75^{\circ}$
Option 1: $1$
Option 2: $4$
Option 3: $3$
Option 4: $2$
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Correct Answer: $3$
Solution :
$\frac{\left (\tan20^{\circ} \right )^{2}}{\left (\operatorname{cosec70^{\circ}} \right )^{2}}+\frac{\left (\cot20^{\circ} \right )^{2}}{\left (\sec70^{\circ} \right )^{2}}+2\tan15^{\circ} \tan45^{\circ} \tan75^{\circ}$
$=\frac{\left (\tan(90^\circ-70^{\circ}) \right )^{2}}{\left (\operatorname{cosec70^{\circ}} \right )^{2}}+\frac{\left (\cot(90^\circ-70^{\circ} )\right )^{2}}{\left (\sec70^{\circ} \right )^{2}}+2\tan(90^\circ-75^{\circ}) \tan45^{\circ} \tan75^{\circ}$
$=\frac{\left (\cot 70^{\circ} \right )^{2}}{\left (\operatorname{cosec70^{\circ}} \right )^{2}}+\frac{\left (\tan 70^{\circ} \right )^{2}}{\left (\sec70^{\circ} \right )^{2}}+2\cot 75^{\circ} \tan45^{\circ} \tan75^{\circ}$
$=\frac{\frac{\cos^270^\circ}{\sin^270^\circ}}{\frac{1}{\sin^270^\circ}}+\frac{\frac{\sin^270^\circ}{\cos^270^\circ}}{\frac{1}{\cos^270^\circ}}+2×\frac{1}{\tan75^\circ}×1×\tan75^{\circ}$
$=\cos^270^\circ+\sin^270^\circ+2$
$=1+2$
$=3$
Hence, the correct answer is $3$.
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