To be honest, I trusted an online Russian-English translator that suggested both terms of English analogues of that notion, specifying that the term "symmetric" is applied most commonly to the notion of function and that indeed seemed rather substantiated as it directly corresponds to the graphic interpretation as cosine graph y = cos x is symmetric in respect of 0y axis while the word "even" - as I was able to draw- is used mostly in conjunction with numbers.
"you have ignored the answers when theta = (pi +- pi/6 ) +- 2pi" - this is totally unclear to me and seems quite "overloaded". How did the expression (pi +- pi/6 ) +- 2pi appear? The common formula for the roots of tangential equation is x = arctan y + pi n, where pi is a minimal positive period of tan. Further on, as cosine is symmetric or, if you prefer more, "even" then whether the argument positive or negative, the cosine is always positive. The value of theta was found via transformation to homogeneous via transformation in turn to tangential equation. It had two solutions - positive and negative. sec theta = 1/cos theta. As it follows from above, cos theta is always positive. Hence, the final result in this case is also positive. You preferred to find the roots via sinusoidal equation. So to get cos theta you as I suppose applied the main trigonometric identity, where it's not important if sine is positive or negative. You may assert that the root has two signes, but there's a question: is there indication of what a quadrant of Cartesian coordinate system should be considered? No, and it means both values must be considered and thus I logically came up with denial of myself You were right, but how it happened that I missed that solution? Mine looks formally correct.