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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 A2 = √1−cosA1+cosA

tan 22½° = √1−cos45°1+cos45°

tan 22½° = √1−1√21+1√2, [Since we know that cos 45° = 1√2]

tan 22½° = √√2−1√2+1

tan 22½° = √√2−1√2+1⋅√2−1√2−1

tan 22½° = √(√2−1)22−1

tan 22½° = √2 - 1

Therefore, tan 22½° = √2 - 1

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11 and 12 Grade Math

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