Question

if sina + 2cosa = 1
then prove 2sina - cosa = 2

Answer 1

sin x+2 cos x =1 (given)

we can write this as: sin x +cos x= 1- cos x

when we square on both sides we get:

sin ^2 x +cos^x+2 sin x cos x=1+cos^2 x-2 cos x ,on simplifying we get

2 cos x(sin x +1)=cos^2 x

2 sin x +2= cos x

2 sin x-cos x =-2

Answer 2

Given SinA + 2CosA=1
To prove 2SinA-CosA=2

Proof:-

We can rewrite SinA+2CosA-1=0
Now multiply the whole equation by SinA - 2CosA +1
we get.....
(SinA+2CosA-1)(SinA-2CosA+1)=0
With the identity a^2-b^2 we get
Sin^2A + 4Cos^2A -1 =0
As we know that 1-cos^2 A = Sin^2A
Therefore we get

1-Cos^2A + 4Cos^2A -1 =0
Therefore we get,,,,
5cos^2A=0
So, Cos^2A=0
Therefore taking square root
CosA=0
Therefore Cos 90=0
Now put this value in given equation (1st equation)
SinA + 0 =1
Therefore SinA=1
Now put these values in proving equation we get
2sinA-CosA=2
2(1)-0 = 2-0 =2
LHS=RHS


Hope you are helped.....