Question

-1 1213156.cos+ sin-12 = sin-165

Answer 1

We have to prove that cos −1 12 13 + sin −1 3 5 = sin −1 56 65 .

Consider sin −1 3 5 =x, then,

sinx= 3 5 cosx= 1− sin 2 x = 1− ( 3 5 ) 2 = 4 5

Another trigonometric function is,

tanx= sinx cosx = 3 4 x= tan −1 3 4 sin −1 3 5 = tan −1 3 4

Consider cos −1 12 13 =y, then,

cosy= 12 13 siny= 1− cos 2 y = 1− ( 12 13 ) 2 = 5 13

Another trigonometric function is,

tany= siny cosy = 5 12 y= tan −1 5 12 cos −1 12 13 = tan −1 5 12

Substitute sin −1 3 5 = tan −1 3 4 and cos −1 12 13 = tan −1 5 12 to left hand side of the given equation,

cos −1 12 13 + sin −1 3 5 = tan −1 5 12 + tan −1 3 4 = tan −1 ( 3 4 + 5 12 1− 3 4 × 5 12 ) = tan −1 36+20 48−15 = tan −1 56 33

Consider the right hand side sin −1 56 65 =z, then,

sinz= 56 65 cosz= 1− sin 2 z = 1− ( 56 65 ) 2 = 33 65

Another trigonometric function is,

tanz= sinz cosz = 56 33 z= tan −1 56 33 sin −1 56 65 = tan −1 56 33

Hence, it is proved that L.H.S.=R.H.S..