Question

A semicircle is described on 'XY' as diameter and points A,B,C and D are chosen on  arc 'XY' and quadrilateral ABCD is drawn.Then maximise tanA+tanB+tanC+tanD.

Answer 1

the triangles DAC and DBC  are right angled triangle
so C+D=90 =90-D
for a quadrilateral
A+B+C+D=360
A+B=270
tan A + tan B + tan C + tan D=tan A + tan (270-A) + tan (90-D) + tan D
=tan A + cot A + cot D + tan D

Answer 2

The triangles DAC and DBC  are right angled triangle
so C+D=90 =>C=90-D
for a quadrilateral
A+B+C+D=360 =>A+B+90=360=>A+B=360-90
A+B=270
tan A + tan B + tan C + tan D=tan A + tan (270-A) + tan (90-D) + tan D
=tan A + cot A + cot D + tan D