Calculus help! please?

Question

Express cos(2arcsinx) without the use of trigonometric or inverse trig functions.

Assume rcos(theta)=x and rsin(theta)=y. Express the equation in a simple form in terms of x and y.

r=3cos(theta)

Please explain how to do these i am so confused

Answer 1

cos( 2 arcsin x)

arcsin x is an angle whose sin is x, let's call it A.

draw a right triangle, label one acute angle A, side opposite x, hypotenuse 1.  so sin A = x/1 = x

by Pythagoras, cos A = √(1-x²).

so we want cos( 2A) = cos² A - sin² A = (1-x²) - x² = 1 - 2x²

r = 3 cos Θ, so r² = 3r cos Θ = 3x

but r² = x² + y², so we have

x² + y² = 3x

x² - 3x + y² = 0

x² - 3x + 9/4 + y² = 9/4

(x - 3/2)² + y² = 9/4

that's a circle, center at (3/2, 0), radius 3/2