Write sin(sin^-1x-cos^-1x) as an algebraic expression in x.

Question

This is for my trig class, thanks for the help!

Answer 1

sin(a-b)=sinacosb-cosasinb

a=arcsinx   b=arccosx

sin(arcsinx)cos(arccosx)-cos(arcsinx)sin(arccosx)

sin(arcsinx)=x

cos(arccosx)=x

cos(arcsinx)=√(1-x²)

sin(arccosx)=√(1-x²)

x²-(1-x²)=2x²-1

 

Answer 2

Sin ( Sin ^-1 X -  Cos ^-1 X ) =

 

 

Sin ( Sin^-1 X )  - Sin ( Cos ^-1 X ) =

 

 

      Sin(sin^-1 X ) = X

      Sin ( Cos^-1X ) = √(1 - X²)

 

      Sin ( A - B) = SinACosB -Sin BCOSA

 

 Sin ( Sin^-1X - Cos ^-1 X ) = X.X - √(1 - X²).√(1- X²) =

                                 

                                  = X² -(1 -X²)=

                                 

                                 = 2X² - 1