Exact Value of tan 22½°

How to find the exact value of tan 22½° using the value of cos 45°?

Solution:

22½° lies in the first quadrant.

Therefore, tan 22½° is positive.

For all positive values of the angle A we know that, tan $$\frac{A}{2}$$ = $$\sqrt{\frac{1 - cos A}{1 + cos A}}$$

tan 22½° = $$\sqrt{\frac{1 - cos 45°}{1 + cos 45°}}$$

tan 22½° = $$\sqrt{\frac{1 - \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}}}$$, [Since we know that cos 45° = $$\frac{1}{\sqrt{2}}$$]

tan 22½° = $$\sqrt{\frac{\sqrt{2} - 1}{\sqrt{2} + 1}}$$

tan 22½° = $$\sqrt{\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1}}$$

tan 22½° = $$\sqrt{\frac{(\sqrt{2} - 1)^{2}}{2 - 1}}$$

tan 22½° = √2 - 1

Therefore, tan 22½° = √2 - 1

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