Question

Find the value of $\tan \left(22{\frac{1}{2}}^{{}^{\circ }}\right)$

• A. $\sqrt{2}-1$
• B. $\sqrt{2}+1$
• C. $\sqrt{3}-1$
• D. $\sqrt{3}+1$

Answer 1

The correct option is A $\sqrt{2}-1$

We know that $\tan \frac{A}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}$
Also, $22{\frac{1}{2}}^o=\frac{{45}^o}{2}$
Since the angle is in first quadrant, tangent of the angle will be positive.
So, $tan\left(22{\frac{1}{2}}^o\right)=tan\left(\frac{{45}^o}{2}\right)=\sqrt{\frac{1-\cos {45}^o}{1+\cos \left({45}^o\right)}}=\sqrt{\frac{1-\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}}$
$=\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}=\sqrt{\frac{(\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}+1)(\sqrt{2}-1)}}=\sqrt{\frac{(\sqrt{2}-1)(\sqrt{2}-1)}{(2-1)}}=(\sqrt{2}-1)$