I was given a task to prove: sec ⁡ θ sec ⁡ θ tan ⁡ θ = sec ⁡ θ ( sec ⁡ θ + tan ⁡ θ )Then I replaced them with their Ratio and Reciprocal Identities sec ⁡ θ = 1 cos ⁡ θ tan ⁡ θ = sin ⁡ θ cos ⁡ θ so I came up with this: 1 cos ⁡ θ 1 cos ⁡ θ ⋅ sin ⁡ θ cos ⁡ θ = 1 cos ⁡ θ ( 1 cos ⁡ θ + sin ⁡ θ cos ⁡ θ ) then, 1 cos ⁡ θ sin ⁡ θ cos ⁡ θ = 1 cos ⁡ θ ( 1 + sin ⁡ θ cos ⁡ θ ) and I had the reciprocal, 1 cos ⁡ θ ⋅ cos ⁡ θ sin ⁡ θ = 1 cos ⁡ θ ( 1 + sin ⁡ θ cos ⁡ θ ) and I don't know what to do next, can someone explain to me how?

Question

I was given a task to prove:            sec      ⁡      θ              sec      ⁡      θ      tan      ⁡      θ        =  sec  ⁡  θ  (  sec  ⁡  θ  +  tan  ⁡  θ  )Then I replaced them with their Ratio and Reciprocal Identities                    sec        ⁡        θ                            =                  1                      cos            ⁡            θ                                              tan        ⁡        θ                            =                              sin            ⁡            θ                                cos            ⁡            θ                              so I came up with this:            1              cos        ⁡        θ                            1                  cos          ⁡          θ                    ⋅                        sin          ⁡          θ                          cos          ⁡          θ                      =      1          cos      ⁡      θ            (          1              cos        ⁡        θ              +                  sin        ⁡        θ                    cos        ⁡        θ              )  then,            1              cos        ⁡        θ                            sin        ⁡        θ                    cos        ⁡        θ              =      1          cos      ⁡      θ            (    1    +                  sin        ⁡        θ                    cos        ⁡        θ              )  and I had the reciprocal,      1          cos      ⁡      θ        ⋅            cos      ⁡      θ              sin      ⁡      θ        =      1          cos      ⁡      θ            (    1    +                  sin        ⁡        θ                    cos        ⁡        θ              )  and I don&#039;t know what to do next, can someone explain to me how?

Josie Stephenson · Accepted Answer

It&#039;s wrong.Here&#039;s why:  LHS  =            sec      ⁡      θ              sec      ⁡      θ      tan      ⁡      θ        =      1          tan      ⁡      θ        =      adjacent    opposite              .      .      .      .                          RHS                            =        sec        ⁡        θ        (        sec        ⁡        θ        +        tan        ⁡        θ        )                                          =                  1                      cos            ⁡            θ                                    (                      1                          cos              ⁡              θ                                +                                    sin              ⁡              θ                                      cos              ⁡              θ                                )                                                  =                  1                      cos            ⁡            θ                                    (                                    1              +              sin              ⁡              θ                                      cos              ⁡              θ                                )                                                  =                              1            +            sin            ⁡            θ                                              cos              2                        ⁡            θ                          =                              hypotenuse            +            opposite                    adjacent                    Now, for adjacentopposite to be equal to hypotenuse+oppositeadjacent, ‘‘opposite&quot; must be equal to ‘‘adjacent&quot; and ‘‘hypotenuse&quot; must be 0. Since this cannot be true, I suppose it is wrong

I was given a task to prove: sec ⁡ θ

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