If sin^2x = a and cos^2x=b, then for x in the first quadrant, sin2x+cos2x=?

Answer 1

Hint: Adapting to my notation, s2 + c2 = 2sc + c² - s² = 2√(ab) + b - a ...

And also, recall here that a+b = 1 ... can you simplify all that further?

You can rewrite it all solely with either the variable "a" or "b". Your choice!

Answer 2

So  sin(x) = sqrt(a) and cos(x) = sqrt(b)

cos(2x) = cos^2(x) - sin^2(x) = b - a

sin(2x) = 2sin(x)cos(x) = 2*sqrt(a*b) so

sin(2x) + cos(2x) = 2*sqrt(a*b)  + b - a