Question

If \(A = 30^\circ\) then the value of \(\frac{2 \tan A}{1 - \tan^2 A}\) is

• A. \(2 \tan 30^\circ\)
• B. \(\tan 60^\circ\)
• C. \(2 \tan 60^\circ\)
• D. \(\tan 30^\circ\)

Hint:

Recognizing the double angle formulas for \(\sin(2A)\), \(\cos(2A)\), and \(\tan(2A)\) can turn a multi-step calculation into a single-step problem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The given expression is the formula for the tangent of a double angle. Recognizing this identity is the quickest way to solve the problem.

Step 2: Key Formula or Approach:
The double angle identity for tangent is:
\[ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} \]

Step 3: Detailed Explanation:
The expression given is \(\frac{2 \tan A}{1 - \tan^2 A}\).
Using the double angle identity, this is equal to \(\tan(2A)\).
We are given that \(A = 30^\circ\).
Substitute this value into \(\tan(2A)\):
\[ \tan(2 \times 30^\circ) = \tan(60^\circ) \] Alternatively, without using the identity:
We know \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
\[ \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} = \frac{2 \left(\frac{1}{\sqrt{3}}\right)}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \frac{\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}} = \frac{2}{\sqrt{3}} \times \frac{3}{2} = \frac{3}{\sqrt{3}} = \sqrt{3} \] And we know that \(\tan 60^\circ = \sqrt{3}\). Both methods give the same result.

Step 4: Final Answer:
The value of the expression is equal to \(\tan 60^\circ\).