I need help with proving the identity cos²x/(1-sinx)² = (secx+tanx)². thanks!?

Answer 1

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

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

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

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

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

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

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

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

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

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

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

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

(1 + sin(x))/(1 - sin(x)) = (1 + sin(x))/(1 - sin(x))

Answer 2

Try turning the right side into the left by first converting everything into sines and cosines.