Question

1.is sin C = cos A in a right angled isosceles triangle ABC where angleB = 90 degrees explain with reason
2.D and E are two points on side AB and BC of a ABC is DE // AC if DB=4,DA=2,BE=6,EC=3 ? explain with proper reason

Answer 1

1. If the right angle triangle ABC, right angled at B is isosceles
See the attachment. 

In triangle ABC,
sinC = AB/AC -----------(1)
cosA = AB/AC -----------(2)
From (1) and (2)
sinC = cosA

2. Given that DE||AC
DB=4,DA=2,BE=6, and we need to check if EC=3.
Let EC = x.
See attachment for diagram.

by basic proportionality theorem,
BD/DA = BE/EC
⇒ 4/2 = 6/x
⇒ 2 = 6/x
⇒ x = 6/2
⇒ x = 3

Thus, it is proved that EC=3.