How do I write son (cos^-1u-tan^-1v) as a algebraic expression

Question

What is the answer and how did you get it

Answer 1

Let θ = cos^-1u.   Then cosθ = u

 

   Draw a right triangle with an acute angle θ.  Label the side adjacent to θ as u and label the hypotenuse as 1.  By the Pythagorean Theorem, the side opposite to θ is √(1-u²).

 

Let β = tan^-1v.  Then tanβ = v

 

  Draw a right triangle with an acute angle β.  Label the side opposite to β as v and the side adjacent to β as 1.  Then, by the Pythagorean Theorem, the hypotenuse is √(1+v²).

 

So, sin(cos^-1u + tan^-1v) = sin(θ + β)

 

                                  = sinθcosβ + cosθsinβ

 

                                  =  (√(1-u²))(1/√(1+v²)) + u(v/√(1+v²))

 

                                  = [uv + √(1-u²)]/√(1+v²)