This could be tan(x)+sec2(x)=1 or tan(x)+sec(2x)=1.
(1) tan(x)+sec2(x)=1
tan2(x)=sec2(x)-1, so tan(x)+tan2(x)=0=tan(x)(1+tan(x)).
So tan(x)=0⇒x=0 or tan(x)=-1⇒x=-¼π.
(2) tan(x)+sec(2x)=1
tan(x)+1/(2cos2(x)-1)=1,
tan(x)+1/(2/sec2(x)-1)=1,
tan(x)+1/(2/(1+tan2(x))-1)=1,
tan(x)+(1+tan2(x))/(1-tan2(x))=1,
tan(x)-tan3(x)+1+tan2(x)=1-tan2(x),
tan(x)-tan3(x)+2tan2(x)=0,
-tan(x)+tan3(x)-2tan2(x)=0,
tan(x)(tan2(x)-2tan(x)-1)=0, so tan(x)=0⇒x=0, or:
tan2(x)-2tan(x)=1,
tan2(x)-2tan(x)+1=2,
(tan(x)-1)2=2,
tan(x)=1±√2⇒x=⅜π (67.5°) or -⅛π (-22.5°).