Showing that arctan ⁡ ( cos ⁡ ( α ) ∗ tan ⁡ ( x ) ) = arccos ⁡ ( cos ⁡ ( x ) cos ⁡ ( arcsin ⁡ ( sin ⁡ ( α ) ∗ sin ⁡ ( x ) ) ) )

Question

Showing that   arctan  ⁡  (  cos  ⁡  (  α  )  ∗  tan  ⁡  (  x  )  )  =  arccos  ⁡  (                    cos        ⁡        (        x        )                    cos        ⁡        (        arcsin        ⁡        (        sin        ⁡        (        α        )        ∗        sin        ⁡        (        x        )        )        )              )

Miah Carlson · Accepted Answer

Notice that   cos  ⁡  (  arcsin  ⁡  (  x  )  )  =      1    −          x      2       by drawing a triangle where   sin  ⁡  θ  =  xSimilarly,   tan  ⁡  (  arccos  ⁡  (  x  )  )  =            1      −              x        2              x    =            1              x        2              −    1  Now                    tan        ⁡        (        R        H        S        )                            =        tan        ⁡        arccos        ⁡                              (                          cos        ⁡        x        ⋅                  1                      1            −            (            sin            ⁡            α            sin            ⁡            x                          )              2                                                            )                                                            =                              1                                                            (                                            cos              ⁡              x              ⋅                              1                                  1                  −                  (                  sin                  ⁡                  α                  sin                  ⁡                  x                                      )                    2                                                                                                                    )                                                  2                                              −          1                                                  =                                            1              −                              sin                2                            ⁡              α                              sin                2                            ⁡              x                                                      cos                2                            ⁡              x                                −          1                                                  =                                                            sin                2                            ⁡              x              (              1              −                              sin                2                            ⁡              α              )                                                      cos                2                            ⁡              x                                                                        =        tan        ⁡        x        cos        ⁡        α            as desired.

Showing that arctan(cos(alpha) tan(x))=arccos(cos(x) cos(arcsin(sin(alpha) sin(x))))

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