Answer to Question #129212 in Trigonometry for venkat

Question

1. Prove that ( 1 - cos² x ) cosec² x = 1 .

2. Prove that (tan x sin x ) + cos x = sec x .

3. Prove that Tan⁴ x + Tan² x = sec⁴ x - sec² x .

4. prove that Cot x + Tan x = sec x cosec x.

5. Prove that (1 + sin x) (1 - sin x) = (sec x - Tan x)² .

Answer 1

1.      Prove that (1- cos² x) cosec ² x =1.

1 – cos²x = sin²x

                                = Sin² x cosec ² x

                                            Cosec ² x= 1/sin² x

                                = sin² x . 1/sin² x

                                 =1         Proved

2.      Prove that (tan x sin x ) + cos x = sec x .

          Tan x =sin x/cos x

                     = (Sin x/co x)sin x + cos x

                         = Sin² x/co x + cos x

                      = (sin² x + cos² x)/cos x

But sin² x +cos² x =1 

    Then;                   1/cos x =sec x       hence proved

3. Prove that Tan ⁴ x + Tan² x = sec⁴ x - sec² x

                         Tan⁴ x +Tan ² x=Tan²x(Tan² x + 1)

                            But    Tan² x = sin² x/cos² x

                                                   =sin² x/cos² x(sin² x/cos² x + 1)

                                                    =sin² x/cos² x {( sin² x+cos² x)/cos² x}

                                                    =sin² x/cos² x .1/cos² x

                                                    =sin² x/cos⁴ x

                Sin²x= 1- cos ² x

                                                    = 1-cos² x/cos⁴ x

                                                     =1/cos ⁴ x – cos² x/cos⁴ x

                                                      =1/cos⁴x - 1/cos ² x       where 1/cos x = sec x  

                                                      Sec⁴ x – sec² x      proved.

4.Prove that Cot x + Tan x = sec x cosec x.  

                 Cot x + Tan x =cos x/ sin x + sin x/cos x

                                         = (sin² x + cos² x)/(sin x cos x)

                                         = 1/ (sin x cos x) =1/sin x .1/cos x

Where 1/sin x=cosec x

              1/cos x=sec x

                 1/cos x .1/sin x =sec x cosec x           Proved

5.  Prove that (1 + sin x)/(1 - sin x) = (sec x + Tan x)² .

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

                                                                 = (1 + sin x) ²/(1 – sin² x)

                                                                  = (1 + sin x)²/cos²x

= {(1 + sin x)/cos x} ²

           = (1/cos x + sin x/cos x)²

= (sec x + tan x)²         proved