Solve tan 2 ⁡ ( x ) + tan ⁡ ( x ) = 2 for 0 ≤ x ≤ 2 πI am trying to solve the trigonometric equation tan 2 ⁡ ( x ) + tan ⁡ ( x ) = 2for 0 ≤ x ≤ 2 π. At first glance, I try to rearrange the trigonometric equation into something more manageable such that sin 2 ⁡ ( x ) cos 2 ⁡ ( x ) + sin ⁡ ( x ) cos ⁡ ( x ) = 2 sin 2 ⁡ ( x ) + sin ⁡ ( x ) cos ⁡ ( x ) = 2 cos 2 ⁡ ( x ) . I do not see how to progress from here. I have also tried using the Pythagorean identity tan 2 ⁡ ( x ) = sec 2 ⁡ ( x ) − 1, but this did not seem to help. Any suggestions are appreciated.

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Solve       tan    2    ⁡  (  x  )  +  tan  ⁡  (  x  )  =  2 for   0  ≤  x  ≤  2  πI am trying to solve the trigonometric equation      tan    2    ⁡  (  x  )  +  tan  ⁡  (  x  )  =  2for   0  ≤  x  ≤  2  π. At first glance, I try to rearrange the trigonometric equation into something more manageable such that                                                        sin              2                        ⁡            (            x            )                                              cos              2                        ⁡            (            x            )                          +                              sin            ⁡            (            x            )                                cos            ⁡            (            x            )                                              =        2                                      sin          2                ⁡        (        x        )        +        sin        ⁡        (        x        )        cos        ⁡        (        x        )        =                    2                  cos          2                ⁡        (        x        )        .            I do not see how to progress from here. I have also tried using the Pythagorean identity       tan    2    ⁡  (  x  )  =      sec    2    ⁡  (  x  )  −  1, but this did not seem to help. Any suggestions are appreciated.

tektonikafrs · Accepted Answer

Unfortunately, the algebraic manipulations you have tried so far do not help much. When solving trigonometric equations, it helps to be on the lookout for &#039;disguised quadratics&#039;. The equation       tan    2    ⁡  x  +  tan  ⁡  x  =  2 is actually a quadratic equation in   tan  ⁡  x. This might be clearer if we set   y  =  tan  ⁡  x: then we have       y    2    +  y  =  2, which is equivalent to       y    2    +  y  −  2  =  0. From here, you can solve this quadratic equation as normal, which gives you the possible values of   y. Then, by substituting   tan  ⁡  x back in for   y, you can find the possible values of   x in the range   0  ≤  x  ≤  2  π.

Solve tan^2(x)+tan(x)=2 for 0<=x<=2pi

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