Write the trignometric expression as an algebraic expression

Question

cos( Arc tan square root of 3 - Arc Cos x )

Answer 1

cos(arc tan sqrt(3) - arccos(x)) =

cos(arctan sqrt(3)) cos(arccos(x)) + sin(arc tan sqrt(3)) sin(arccos(x))

arctan sqrt(3) = π/3; cos(π/3) = 1/2, and sin(π/3) = √3/2

cos(arccos(x)) = x

sin(arccos(x)) = √(1-x²); since arccos takes on values from 0 to π, sin is non-negative, so we only have the positive square root here.

Therefore, cos(arctan sqrt(3)) cos(arccos(x)) + sin(arc tan sqrt(3)) sin(arccos(x)) =

1/2x + √3/2 * √(1-x²) or 1/2 x + 1/2 √(3-3x² ) Note that the domain of x is [-1,1], as that is the domain for which arccos is defined - cos can only have values between -1 and 1