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Question

Mathematics Question on Trigonometric Functions

The value of tan⁡π8\tan \frac{\pi}{8} is equal to

A

12\frac{1}{2}

B

2+1\sqrt{2} + 1

C

12+1\frac{1}{\sqrt{2} + 1}

D

1−21 - \sqrt{2}

Answer

12+1\frac{1}{\sqrt{2} + 1}

Explanation

Solution

We know that, tan⁡θ=sin⁡ 2θ1+cos⁡ 2θ\tan \theta=\frac{\sin\, 2 \theta}{1+\cos\, 2 \theta}
⇒tan⁡π8=sin⁡ 2(π/8)1+cos⁡ 2(π/8)\Rightarrow \tan \frac{\pi}{8}=\frac{\sin\, 2(\pi / 8)}{1+\cos \,2(\pi / 8)}
⇒tan⁡π8=sin⁡(π/4)1+cos⁡(π/4)\Rightarrow \tan \frac{\pi}{8}=\frac{\sin (\pi / 4)}{1+\cos (\pi / 4)}
∴tan⁡π8=121+12\therefore \tan \frac{\pi}{8}=\frac{\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}
=12+1=\frac{1}{\sqrt{2}+1}