Solve sec ⁡ x + tan ⁡ x = 4Find x for 0 &lt; x &lt; 2 πEventually I get cos ⁡ x = 8 17 x = 61.9 ∘ The answer I obtained is the only answer, another respective value of x in 4-th quadrant does not solve the equation, how does this happen? I have been facing the same problem every time I solved this kind of trigonometric equation.

Question

Solve   sec  ⁡      x    +  tan  ⁡      x    =  4Find x for   0  &amp;lt;  x  &amp;lt;  2  πEventually I get  cos  ⁡  x  =      8    17    x  =      61.9          ∘      The answer I obtained is the only answer, another respective value of x in 4-th quadrant does not solve the equation, how does this happen? I have been facing the same problem every time I solved this kind of trigonometric equation.

marktje28 · Accepted Answer

Using t-formulaLet       t    =    tan    ⁡          x      2      , then       cos    ⁡    x    =                  1        −                  t          2                            1        +                  t          2                     and       tan    ⁡    x    =                  2        t                    1        −                  t          2                    Now                                          1            +                          t              2                                            1            −                          t              2                                      +                              2            t                                1            −                          t              2                                                          =        4                                                  (            1            +            t                          )                              2                                                          1            −                          t              2                                                          =        4                                                  1            +            t                                1            −            t                                              =        4                        (        t        ≠        −        1        )                            t                            =                  3          5                                    tan        ⁡                  x          2                                    =                  3          5                                    x                            =        2                  (          n          π          +                      tan                          −              1                                ⁡                      3            5                    )                                    x                            =        2                  tan                      −            1                          ⁡                  3          5                                (        0        &amp;lt;        x        &amp;lt;        2        π        )

Solve sec ⁡ <mrow class="MJX-TeXAtom-ORD"> x </mrow> + tan

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