Question

1
is
The value of tan 22
2​

Answer 1

22½° lies in the first quadrant.

Therefore, tan 22½° is positive.

For all positive values of the angle A we know that, tan A2A2 = 1−cosA1+cosA−−−−−√1−cosA1+cosA

tan 22½° = 1−cos45°1+cos45°−−−−−−√1−cos45°1+cos45°

tan 22½° = 1−12√1+12√−−−−−√1−121+12, [Since we know that cos 45° = 12√12]

tan 22½° = 2√−12√+1−−−−√2−12+1

tan 22½° = 2√−12√+1⋅2√−12√−1−−−−−−−−−−√2−12+1⋅2−12−1

tan 22½° = (2√−1)22−1−−−−−−√(2−1)22−1

tan 22½° = √2 - 1

Therefore, tan 22½° = √2 - 1