Question: solve cos2x+sin2x=0 .
cos2x+sin2x=0 --- rearrange
cos2x = -sin2x --- divide both sides by cos2x
1 = -sin2x/tan2x, i.e.
tan2x = -1
Now tan(α) is positive if α is in the 1st quadrant or the 3rd quadrant.
But, since we have tan2x as being negative then 2x must lie in the 2nd quadrant or the 4th quadrant.
Since tan(-α) = -tan(α) and tan(π/4) = 1, then tan(-π/4) = -tan(π/4) = -1
i.e. 2x = -π/4, when 2x is in the 4th qaudrant
and 2x = 3π/4, when 2x is in the 3rd quadrant
Answer: x = -π/8, x = 3π/8