Write the given expression as an algebraic expression in x c

Write the given expression as an algebraic expression in x. cos(2 tan^-1(x)) Write the given expression as an algebraic expression in x. cos(2 cos^-1(x)) Find the exact value of the given expression. sin(2 cos^-1(4/5))

Solution

1) cos ( 2 tan^-1 (x) )

let tan^-1 x = y

tan y = x

tan theta = perpendicular / base = x/ 1

hypotenuse = sqrt (x^2 + 1)

therefore,

sin y = perpendicular / hypotenuse = x / sqrt (x^2 + 1)

cos y = base / hypotenuse = 1 / sqrt (x^2 + 1)

so , cos (2y ) = cos^2y - sin^2 y = 1 / x^2 + 1 - x^2 / x^2 + 1

cos (2 (tan^-1 x) ) = (1-x^2) / (x^2 + 1)

2)   cos ( 2 cos^-1 (x) )

let cos^-1 x = y

cos y = x = base / hypotenuse

perpendicular = sqrt (1- x^2 )

sin y = sqrt (1-x^2) / x

cos (2y) = cos^2y - sin^2 y

= x^2 - (1-x^2) / x^2

= (x^4 - 1 + x^2 )/ x^2

cos ( 2 cos^-1 (x) ) = (x^4 + x^2-1 )/ x^2

3) sin (2 cos^-1 (4/5))

let cos^-1 (4/5) = y

cos y = 4/5 = base / hypotenuse

perpendicular = sqrt ( 5^2 - 4^2 ) = 3

sin y = 3 / 5

sin (2 y ) = 2 sin y cos y

= 2 (3/5) ( 4/5) = 24 / 25

hence,

sin ( 2 cos^-1 (4/5)) = 24/25