Q2(i):

Choose the correct option and justify your choice : (i) $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =$

Given: The trigonometric expression $\frac{2 \tan 30^\circ}{1 + \tan^2 30^\circ}$.

To Find: The value of the expression and identify the correct option among the standard trigonometric values.

Step 1: Identify the value of the trigonometric ratio.
From the standard trigonometric table for specific angles:
$\tan 30^\circ = \frac{1}{\sqrt{3}}$

Step 2: Substitute the value into the given expression.
Let the expression be $E$.
$E = \frac{2 \left( \frac{1}{\sqrt{3}} \right)}{1 + \left( \frac{1}{\sqrt{3}} \right)^2}$

Step 3: Simplify the numerator and the denominator.
Numerator: $2 \times \frac{1}{\sqrt{3}} = \frac{2}{\sqrt{3}}$
Denominator: $1 + \left( \frac{1}{\sqrt{3}} \right)^2 = 1 + \frac{1}{3}$
[Since $(\sqrt{a})^2 = a$]
Denominator: $\frac{3}{3} + \frac{1}{3} = \frac{4}{3}$

Step 4: Perform the division of the fractions.
$E = \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}}$
To divide by a fraction, we multiply by its reciprocal:
$E = \frac{2}{\sqrt{3}} \times \frac{3}{4}$
$E = \frac{2 \times 3}{4 \times \sqrt{3}}$
$E = \frac{6}{4\sqrt{3}}$

Step 5: Simplify the resulting fraction.
$E = \frac{3}{2\sqrt{3}}$
Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$:
$E = \frac{3 \times \sqrt{3}}{2 \times \sqrt{3} \times \sqrt{3}}$
$E = \frac{3\sqrt{3}}{2 \times 3}$
$E = \frac{3\sqrt{3}}{6}$
$E = \frac{\sqrt{3}}{2}$

Step 6: Compare with standard trigonometric values.
We know that:
$\sin 60^\circ = \frac{\sqrt{3}}{2}$
$\cos 60^\circ = \frac{1}{2}$
$\tan 60^\circ = \sqrt{3}$
$\sin 30^\circ = \frac{1}{2}$

Since the calculated value is $\frac{\sqrt{3}}{2}$, it corresponds to $\sin 60^\circ$.

Final Answer: The value of the expression is $\frac{\sqrt{3}}{2}$, which corresponds to $\sin 60^\circ$.