If tan ⁡ θ + sin ⁡ θ = m and m 2 − n 2 = 4 m n so prove that tan ⁡ θ − sin ⁡ θ = nwe have to take 4 m ( tan ⁡ θ − sin ⁡ θ ) = 4 m n to satisfy tan ⁡ θ − sin ⁡ θ = n but I don't like it, I just want direct derivation.

Question

If   tan  ⁡  θ  +  sin  ⁡  θ  =  m and       m    2    −      n    2    =  4      m    n   so prove that   tan  ⁡  θ  −  sin  ⁡  θ  =  nwe have to take   4      m    (    tan    ⁡    θ    −    sin    ⁡    θ    )    =  4      m    n   to satisfy   tan  ⁡  θ  −  sin  ⁡  θ  =  n but I don&#039;t like it, I just want direct derivation.

Cseszteq5j · Accepted Answer

Let&#039;s call   x  =  tan  ⁡  θ  −  sin  ⁡  θ then  (  tan  ⁡  θ  −  sin  ⁡  θ      )    2    +  4      m    (    tan    ⁡    θ    −    sin    ⁡    θ    )    −      n    2    −  4      m    n    =  0becomes                              x          2                +        4                  m                          x                −                  n          2                −        4                  m                          n                                    =        0                            (                  x          2                −                  n          2                )        +        4                  m                (                  x                −                  n                )                            =        0                            (        x        −        n        )        (        x        +        n        )        +        4                  m                (                  x                −                  n                )                            =        0                            (                  x                −                  n                )        (                  x                +                  n                )        (        x        +        n        )        +        4                  m                (                  x                −                  n                )                            =        0                            (                  x                −                  n                )                              (                          (                  x                +                  n                )        (        x        +        n        )        +        4                  m                                      )                                              =        0            and since the second term is strictly positive       x    =      n      ⟺    x  =  n as we wanted.

If tan theta+sin theta=m and m^2−n^2=4 sqrt(mn) so prove that tan theta−sin theta=n

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