Question

tan(180/16)=√(4+2√2)-(√2+1)

Answer 1

tan( 180/16)° = tan( 45/2)°

here we have to use important formula ,
tan∅ = √{ (1 - cos2∅)/(1 + cos2∅)}
Let ∅ = (45/2)°
then,
2∅ = 45° , now use formula and find tan(45/2)°

so,
tan(45/2)° = √{(1 - cos2(45/2)°)/(1 + cos2(45/2))}
= √{(1 - cos45)/(1 + cos45)}
= √{(1 - 1/√2)/(1 + 1/√2)}
= √{(√2 - 1)/(√2 + 1)}
now rationalize
= √{( √2-1)(√2 -1)/(√2 + 1)(√2 -1)}
= √{(√2-1)²/(√2²-1²)}
= √2 - 1

hence, tan(45/2)° = √2 -1
hence, tan(180/16)° = √2 -1