Question

If $\tan \alpha = \frac{1}{2}$, then the value of $\tan^2(2\alpha) \sec^2(2\alpha)$ is equal to

• A. $\frac{200}{9}$
• B. $\frac{400}{9}$
• C. $\frac{200}{81}$
• D. $\frac{400}{81}$
• E. $\frac{200}{27}$

Hint:

Always simplify the trigonometric expression into a single ratio (like $\tan \theta$) if possible before substituting numerical values to reduce calculation steps.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given \(\tan \alpha\) and need to evaluate an expression involving the double angle \(2\alpha\).
Step 2: Key Formula or Approach:
Use the double angle formula for tangent:
\[ \tan(2\alpha) = \frac{2\tan \alpha}{1 - \tan^2 \alpha} \] Use the identity: \(\sec^2(2\alpha) = 1 + \tan^2(2\alpha)\).
Compute \(\tan(2\alpha)\) first, then substitute it into the full expression.
Step 3: Detailed Explanation:
Given \(\tan \alpha = \frac{1}{2}\).
Calculate \(\tan(2\alpha)\):
\[ \tan(2\alpha) = \frac{2(1/2)}{1 - (1/2)^2} = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3} \] Now find \(\sec^2(2\alpha)\):
\[ \sec^2(2\alpha) = 1 + \tan^2(2\alpha) = 1 + \left(\frac{4}{3}\right)^2 = 1 + \frac{16}{9} = \frac{25}{9} \] We need to evaluate \(\tan^2(2\alpha)\sec^2(2\alpha)\):
\[ \tan^2(2\alpha) = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \] \[ \tan^2(2\alpha)\sec^2(2\alpha) = \frac{16}{9} \times \frac{25}{9} = \frac{400}{81} \] Step 4: Final Answer:
The value is \(\frac{400}{81}\).