please, prove that 1-2cos^2x = tan^2x-1/tan^2x+1

Question

The worksheet deals withb trigonometric identities and we have to prove each of them. I'm stumped on this one. Please, if you could help, I'd highly appreciate it!

Answer 1

1-2cos²x = 1-2/sec²x

 

                =  sec²x-2

                      sec²x

                =1+tan²x-2       (as 1+tan²x=sec²x)

                   1+tan²x

                = tan²x-1

                   tan²x+1  

Answer 2

 1 - 2 Cos^2 X                                              ( Tan^ 2X -1) / (1 + Tan^2X )=

 

                                                                    ( Sin^2 X - 1 ) /  (Sin^2X +1 )

                                                                      Cos^2 X            Cos^2X 

 

                                                                     ( Sin^2 X - Cos^2 X) * Cos^2X              

                                                                                Cos^2 X           Sin^2X + Cos^2X

 

                                                                           Sin ^2 X - Cos ^2 X =

                                                                       

                                                                        1 - Cos^2 X - Cos ^2 X =

                                                                        

        1 - 2 C0s ^2X                                       =        1 - 2 Cos^2 X