Question

$tan(2tan^{-1}(\frac{2}5))$ is equal to

• A. $\frac8{5}$
• B. $\frac{10}{21}$
• C. $\frac{20}{21}$
• D. $\frac{21}{25}$
• E. $\frac{4}{25}$

Answer 1

The Correct Option is C

Given expression: 

\(\tan \left( 2 \tan^{-1} \left( \frac{2}{5} \right) \right)\)

Using the identity:

\(\tan (2x) = \frac{2 \tan x}{1 - \tan^2 x}\)

Let \( x = \tan^{-1} \left( \frac{2}{5} \right) \), so \(\tan x = \frac{2}{5}\).

Applying the identity:

\(\tan (2x) = \frac{2 \times \frac{2}{5}}{1 - \left(\frac{2}{5}\right)^2}\)

Calculating the denominator:

\(1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}\)

Calculating the numerator:

\(2 \times \frac{2}{5} = \frac{4}{5}\)

Dividing:

\(\frac{\frac{4}{5}}{\frac{21}{25}}\)

\(= \frac{4}{5} \times \frac{25}{21} = \frac{4 \times 25}{5 \times 21} = \frac{100}{105} = \frac{20}{21}\)

Thus, the correct answer is:

\(\frac{20}{21}\)