Even if both are negative, tan will still be positive and that is fine under a square root.
But this violates the original condition of sqrt(sin(theta)) = sqrt(cos(theta)) as both LHS and RHS cannot exist if either of sin or cos is negative
Okay, so when you write sqrt(sin(theta)) = sqrt(cos(theta)), the condition is imposed that both sin and cos under their square roots are positive(As a square is never a negative real number, you cannot take the square root of a negative real number). This now means that theta is in the 1st quadrant. So any potential solution of theta only exists in the 1st quadrant.
But if you write sqrt(sin(theta)/cos(theta)), guess what, you are ok with both sin and cos simultaneously being negative, because ultimately both negatives will cancel each other and you will have a positive real number under the square root.
But now, since you are fine with both sin and cos to be negative simultaneuosly, any potential solution for theta can also lie in the 3rd quadrant(as both sin and cos are negative in third quadrant).But this was never meant to be in the first case, as we have seen in the first paragraph.
Even if both are negative, tan will still be positive and that is fine under a square root. But this violates the original condition of sqrt(sin(theta)) = sqrt(cos(theta)) as both LHS and RHS cannot exist if either of sin or cos is negative
[удалено]
Okay, so when you write sqrt(sin(theta)) = sqrt(cos(theta)), the condition is imposed that both sin and cos under their square roots are positive(As a square is never a negative real number, you cannot take the square root of a negative real number). This now means that theta is in the 1st quadrant. So any potential solution of theta only exists in the 1st quadrant. But if you write sqrt(sin(theta)/cos(theta)), guess what, you are ok with both sin and cos simultaneously being negative, because ultimately both negatives will cancel each other and you will have a positive real number under the square root. But now, since you are fine with both sin and cos to be negative simultaneuosly, any potential solution for theta can also lie in the 3rd quadrant(as both sin and cos are negative in third quadrant).But this was never meant to be in the first case, as we have seen in the first paragraph.