trigonometry!!!!!!!!!!!!!!!!!!!!!

Question

prove that    tan (x/2) + cot (x/2) = 2 csc x

Answer 1

tan(x/2) + cot(x/2) = (sin(x/2) / cos(x/2)) + (cos(x/2) / sin(x/2)) [Since tan(x/2) = sinx/cosx,

                                                                                                           cot(x/2) = cosx/sinx]

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

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

                             =1 / (cos(x/2)sin(x/2))     [ Since sin^2x + cos^2x = 1 ]

                             = 2 / (2cos(x/2)sin(x/2))

                             = 2 / sin(x)                      [ Since 2cos(x/2)sin(x/2) = sinx ]

                             = 2csc(x)                        [ Since 1/sinx = cosecx ]

Hence it is proved that tan(x/2) + cot(x/2) = 2csc(x).