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 first find $\tan 2\alpha$ using the double-angle formula, then use the trigonometric identity $\sec^2 \theta = 1 + \tan^2 \theta$.
Key Formula or Approach:
\[ \tan 2\alpha = \frac{2\tan \alpha}{1 - \tan^2 \alpha} \]

Step 2: Detailed Explanation:
1. Calculate $\tan 2\alpha$:
\[ \tan 2\alpha = \frac{2(1/2)}{1 - (1/2)^2} = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3} \]
2. Calculate $\tan^2 2\alpha$:
\[ \tan^2 2\alpha = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
3. Calculate $\sec^2 2\alpha$:
\[ \sec^2 2\alpha = 1 + \tan^2 2\alpha = 1 + \frac{16}{9} = \frac{9+16}{9} = \frac{25}{9} \]
4. Multiply the two values:
\[ \tan^2 2\alpha \cdot \sec^2 2\alpha = \frac{16}{9} \cdot \frac{25}{9} = \frac{400}{81} \]

Step 3: Final Answer:
The value is $\frac{400}{81}$.