Question

Evaluate: \[ \tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha + 8 \cot 8\alpha. =\]

• A. \( \sin \alpha \)
• B. \( \cos \alpha \)
• C. \( \tan \alpha \)
• D. \( \cot \alpha \)

Hint:

For expressions involving tangent and cotangent sums, look for patterns using trigonometric identities and transformations.

Answer 1

The Correct Option is D

Step 1: Identifying the pattern
The given expression: \[ \tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha + 8 \cot 8\alpha \] follows a pattern involving tangent and cotangent functions. Rewriting \( \cot 8\alpha \) in terms of \( \tan \): \[ \cot 8\alpha = \frac{1}{\tan 8\alpha}. \] Step 2: Using tangent and cotangent properties
The terms can be rewritten using double angle identities: \[ 2 \tan 2\alpha = \frac{2 \sin 2\alpha}{\cos 2\alpha}, \quad 4 \tan 4\alpha = \frac{4 \sin 4\alpha}{\cos 4\alpha}, \quad 8 \cot 8\alpha = \frac{8 \cos 8\alpha}{\sin 8\alpha}. \] Applying trigonometric simplifications leads to: \[ \tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha + 8 \cot 8\alpha = \cot \alpha. \] Step 3: Conclusion
Thus, the final answer is: \[ \boxed{\cot \alpha}. \]