Answer to Question #129212 in Trigonometry for venkat

Question #129212

1. Prove that ( 1 - cosx ) cosecx = 1 .

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

3. Prove that Tanx + 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 .


1
Expert's answer
2020-08-11T15:35:35-0400

1.      Prove that (1- cosx) cosec 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 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




                                   

                        


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