2 tan - 1 tan α 2 tan π 4 - β 2 , α , β ∈ 0 , π 2 is equal to

tan - 1 sin α · cos β cos α + sin β

tan - 1 cos α + sin β sin α · cos β

tan - 1 tan α tan β tan α + tan β

An ordered triplet solution ( x , y , z ) with x , y , z ∈ ( 0 , 1 ) and satisfying x 2 + y 2 + z 2 + 2 x y z = 1 is -

cos π 12 , cos 4 π 9 , cos 17 π 36

cos 2 π 5 , cos π 3 , cos π 10

cos 7 π 12 , cos π 4 , cos π 6

Prove that cos - 1 x = 2 cos - 1 1 + x 2

To prove: cos - 1 x = 2 cos - 1 1 + x 2 Let x = cos 2 θ cos - 1 cos 2 θ = 2 cos - 1 1 + cos 2 θ 2 = 2 cos - 1 2 cos 2 θ 2 = 2 cos - 1 cos θ = 2 θ

Show that sin - 1 3 5 - sin - 1 8 17 = cos - 1 84 85

Let sin − 1 3 5 = x and sin − 1 8 17 = y Therefore sin x = 3 5 and siny = 8 17 Now cos x = 1 − sin 2 x = 1 − 9 25 = 4 5 and cos y = 1 − sin 2 y = 1 − 64 289 = 15 17 We have cos ( x - y ) = cosxcosy + sinxsiny = 4 5 × 15 17 + 3 5 × 8 17 = 84 85 Therefore x − y = cos − 1 84 85 Hence sin − 1 3 5 − sin − 1 8 17 = cos − 1 84 85