i have 2 question -

1)if cos ^_-1 (x/a)+cos^-1(y/b)=d

prove that : x²/a²-2xy/ab-cosd+y²/b²=sin²d

2)solve tan(cos^-1x)=sin(tan^-1x)

3)tan^-1x+tan^-1y+tan^-1z=pi/2

prove that xy+yz+zx =1

 

 cos ^_-1 (x/a)+cos^-1(y/b)=d

cos ^_-1 (x/a) =d - cos^-1(y/b

Taking cos on both sides:

 (x/a)= cos(d - cos^-1(y/b)

(x/a)= cosd cos(cos^-1(y/b) +sind sin(cos^-1(y/b))

x/a= cosd y/b +sind(root of 1- (y/b)²

x/a- y/bcosd= sind(root of 1-(y/b)²                                                            

squaring on both sides,                                                           (sin (cos^{-1_{X = ROOT OF 1- X^2))}}

x²/a²  + y² / b² cos² d -2 xy cosd/ab= sin² d  -  sin² d y²/b²  

x²/a²  + y² / b²⁽ cos² d + sin² d) -2 xy cosd/ab =  sin² d                  (cos² d + sin² d= 1)

x²/a²  + y² / b² -2 xy cosd/ab=  sin² d