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Q)

$\tan 5^{\large\circ}$ is a

$\begin{array}{1 1}(a)\;Rational\;no&(b)\;Irrational\;no\\(c)\;Prime\;no&(d)\;None\;of\;these\end{array}$

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A)
Let $\theta=5^{\large\circ}$
Suppose $\tan 5^{\large\circ}$=Rational no.
Then $6\theta=30^{\large\circ}$
$\tan 6\theta=\tan 30^{\large\circ}$
$\large\frac{6C_1\tan\theta-6C_3\tan^3\theta+6C_5\tan ^5\theta}{1-6C_2\tan^2\theta+6C_4\tan^4\theta-6C_6\tan^6\theta}=\frac{1}{\sqrt 3}$
$\tan n\theta=\large\frac{nC_1\tan\theta-nC_3\tan^3\theta+nC_5\tan^5\theta+....}{1-nC_2\tan^2\theta+nC_4\tan^4\theta-nC_6\tan^6\theta+...}$
$\Rightarrow$ Rational no =Irrational no
$\Rightarrow $ It is impossible
$\Rightarrow$ Our assumption is wrong
Hence (b) is the correct answer.
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