Question

Sinq+2cosq=1 prove 2sinq-cosq=2

Answer 1

sinq+2cosq=1
Squaring both sides,
(sinq+2cosq)²=1²
or, sin²q+2sinq.2cosq+4cos²q=1
or, 4sinqcosq+4cos²q=1-sin²q
or, 4sinqcosq+4(1-sin²q)=cos²q
or, 4sinqcosq+4-4sin²q=cos²q
or, -4sin²q+4sinqcosq-cos²q=-4
or, -{(2sinq)²-2.2sinqcosq+cos²q}=-4
or, (2sinq-cosq)²=2²
or, 2sinq-cosq=2 (Proved)