Question

The value of \( \tan \frac{\pi}{8} \) is

• A. \( \sqrt{2} \)
• B. \( -\sqrt{2} \)
• C. \( \sqrt{2}-1 \)
• D. \( 1-\sqrt{2} \)
• E. \( -1-\sqrt{2} \)

Hint:

Half-angle identities are useful for special angle evaluation.

Answer 1

The Correct Option is C

Concept: Use identity: \[ \tan \frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta} \]

Step 1: Take \( \theta=\frac{\pi}{4} \). \[ \tan\frac{\pi}{8} = \frac{1-\cos\frac{\pi}{4}}{\sin\frac{\pi}{4}} \]

Step 2: Substitute values. \[ \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} \] \[ \tan\frac{\pi}{8} = \frac{1-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} \]

Step 3: Simplify. \[ = \frac{\sqrt{2}-1}{1} = \sqrt{2}-1 \]