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Exact Value of cos 22½°

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

Solution:

22½° lies in the first quadrant.

Therefore, sin 22½° is positive.

For all values of the angle A we know that, cos A = 2 cos2 A2  - 1

⇒ 1 + cos A = 2 cos2 A2

⇒ 2 cos2 A2 = 1 + cos A

⇒ 2 cos2 22½˚ = 1 + cos 45°

⇒ cos2 22½˚ = \frac{1 + cos 45°}{2}

⇒ sin^{2} 22½˚ = \frac{1 + \frac{1}{\sqrt{2}}}{2}, [Since we know cos 45° = \frac{1}{√2}]

⇒ cos 22½˚ = \sqrt{\frac{1}{2}(1 +  \frac{1}{\sqrt{2}})}, [Since, cos 22½˚ > 0]

⇒ cos 22½˚ = \sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}}

⇒ cos 22½˚ = \frac{1}{2}\sqrt{2 + \sqrt{2}}

Therefore, cos 22½˚ = \frac{1}{2}\sqrt{2 + \sqrt{2}}

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