Question

prove that sin-1 3/5+sin-1 8/17=cos-1 36/85

Answer 1

let sin^-1 (3/5) = Q
sinQ =3/5
cosQ = 4/5

sin^-1 (8/17) = B
sinB = 8/17
cos B = 15/17

now ,
Q + B = T (let )
take both side sin
sin (Q + B) = sinT
SinQ .cosB +cosQ.sinB =sinT
put value ,
3/5 x 15/17 + 4/5 x 8/17 = sinT
(45 + 32) /85 = sinT
sinT = 77/85
so,
cosT = { 1- (77/85)^2}^1/2
=36/85
so,
sin^-1 (3/5) +cos^-1 (8/17) =cos^(36/85)
hence proved

Answer 2

Answer:

Let

A=sin^-1 3/5

therefore,sinA=3/5

similarly

B=sin^-1 8/17

sinB=8/17

Now,

sin^2A+Cos^2A=1

therefore,cosA=√1-sin^2A

now put sin A

cosA=15/17

similarly

cosB=4/5

Now if you want to prove it in terms of cos^-1 then use formula Cos(A+B) ,for sin^-1 formula will be sin(A+B)

since we need to prove in terms of cos^-1

therefore,cos(A+B)=cosAcosB-SinASinB

now put values

cos(A+B)=36/85

A+B=cos^-1(36/85)

as ,A=sin^-1 3/5 and B=sin^-1 8/17

therefore,

sin^-1 3/5+sin^-1 8/17=cos^-1 36/85