Recall the definitions:
- sec(x)=1cos(x)
- tan(x)=sin(x)cos(x)
We can thus write sec2(x)+2tan2(x) as
1cos2(x)+2sin2(x)cos2(x)=1+2sin2(x)cos2(x).
We want this expression to equal 4, so we can multiply for cos2(x) and get
1+2sin2(x)cos2(x)=4⇔1+2sin2(x)=4cos2(x).
Subtracting cos2(x) from both sides, we have
1−cos2(x)+2sin2(x)=3cos2(x),
and since 1−cos2(x) equals sin2(x), the expression becomes
3sin2(x)=3cos2(x).
Dividing the whole expression by the right member, we have
3sin2(x)3cos2(x)=1. Canceling the 3's out, and recalling again that tan(x)=sin(x)cos(x), we finally find out that the previous equation is the same as
tan2(x)=1, which is verified (considering only the angles between 0 and 2π) when tan(x)=1, i.e. when x=π4, and when tan(x)=−1, i.e. when x=5π4