prove that cos2x+1/sin2x = sin2x/1-cos2x, show proper steps

Answer 1

(cos2x+1)/sin2x = sin2x/(1-cos2x)

cross multiply

sin2x(sin2x) = (1+cos2x)(1-cos2x) = -(-1+cos2x)(1+cos2x)= -(cos2x-1)(cos2x+1)

sin^2(2x) = -(cos^2(2x)-1)

sin^2(2x) = 1-cos^2(2x)

sin^2(2x) + cos^2(2x) = 1

this is a standard trigonometric identity, sin^2A + cos^2A = 1 for any value of the angle A, including A=2x

draw a unit circle, x^2+y^2=1 is its equation

with center at the origin and radius = 1

x coordinate of a point on the circle =cosA, y coordinate of the same point = sinA

use the Pythagorean Theorem, x^2 + y^2 = 1^2

cos^2A + sin^2A = 1

sin^2A + cos^2A = 1 Let A=2x

sin^2(2x) + cos^2(2x) = 1