Write the following trigonometric expression as an algebraic

Write the following trigonometric expression as an algebraic expression containing u and v. Give the restrictions required on u and v. sin|sin^-1u - cos^-1 v| Write an algebraic expression containing u and v with the restrictions required on u and v. Select the correct choice below and fill in the answer box to complete your choice. sin [sin^-1u - cos^-1v] = uv + Square root 1 + u^2 Square root 1 + v^2, u Inequality v Inequality . sin [sin^-1 u - cos^-1v] = uv + Square root 1 - u^2 Square root 1 - v^2. Inequality u

Solution

sin( sin^-1u +cos^-1v )

use sin(a +b) formula ; sinacosb + cosa sinb

So, sin( sin^-1u +cos^-1v ) = sinsin^-1(u)cos(cos^-1(v) )+ cos(sin^-1(u)sin(cos^-1(v)

   we can write sin^-1u =y

so, sin^-1u = y

u = siny ----> cosy = sqrt( 1-u^2)

y = cos^-1sqrt( 1-u^2)

cos^-1v = z

cosz = v---> sinz = sqrt( 1-v^2)

z = sin^-1(1-v^2)

sin( sin^-1u +cos^-1v ) = sinsin^-1(u)cos(cos^-1(v) )+ cos(sin^-1(u)sin(cos^-1(v)

= u*v + coscos^-1sqrt( 1-u^2)* sin sin^-1(1-v^2)

= u*v +sqrt( 1-u^2)*sqrt( 1- v^2)

Condition : 1- u^2>=0 ---> -1<=u <=1

                       1- v^2>=0 --->   -1 <=v <=1

Option B