Trig Equations Question?

Question

Prove (1-cos(x))/(1+cos(x)) = (csc(x)-cot(x))^2

Answer 1

Right hand side

= (cscx-cotx)^2

     [ (a-b)^2 = a^2+b^2-2ab ]

= csc^2x-2cscxcotx+cot^2x

    [ Recall the formula cscx = 1/sinx and cotx =cosx/sinx ]

= 1/sin^2x - 2*1/sinx*cosx/sinx + cos^2x/sin^2x

    [ Take out common term1/sin^2x ]

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

= 1/sin^2x*(1-cosx)^2

    [ From the trigonometric identities sin^2x =1-cos^2x ]

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

= (1-cosx)(1-cosx)/ (1-cosx)(1+cosx)

   [ Cancel the common term (1-cosx) ]

= (1-cosx)/(1+cosx)

= Left hand side