Question : The value of $\operatorname{cosec}^{2}60°+\sec^{2}60°– \cot^{2}60°+\tan^{2}30°$ will be:
Option 1: $5$
Option 2: $5\frac{1}{2}$
Option 3: $5\frac{1}{3}$
Option 4: $5\frac{2}{3}$
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Correct Answer: $5\frac{1}{3}$
Solution :
Given: $\operatorname{cosec}^{2}60°+\sec^{2}60°– \cot^{2}60°+\tan^{2}30°$
We know that $\operatorname{cosec60°} = \frac{2}{\sqrt{3}}$
$\sec 60° = 2$
$\cot 60° = \frac{1}{\sqrt{3}}$
$\tan 30° = \frac{1}{\sqrt{3}}$
Substituting these values in the given expression, we get:
= $(\frac{2}{\sqrt{3}})^{2} + 2^{2} – (\frac{1}{\sqrt{3}})^{2} +(\frac{1}{\sqrt{3}})^{2}$
= $\frac{4}{3}+4=\frac{16}{3}= 5\frac{1}{3}$
Hence, the correct answer is $5\frac{1}{3}$.
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