Question

prove that 1+2sinA-2cosA/1+2sinA + 2cosA​

Answer 1

Answer:

ANSWER

We know the formula,

sin A = 1-2cosA

so,

sin²A = 1-4cosA+4cos²A

1-cos²A = 1-4cosA+4cos²A

0 = 5cos²A-4cosA

0 = cosA(5cosA-4)

cosA=0 or cosA=4/5

If cosA=0 then one possibility is sinA=1 and in this case 2sinA-cosA=2.

At first I thought cosA=0 and sinA=-1 was possible but this is not consistent with sinA+2cosA=1.

Let's consider the case cosA=4/5.

Then sinA = ±√(1-cos²A) = ±√(1-16/25) = ±3/5

If cosA=4/5 and sinA=3/5 then sinA + 2cosA doesn't equal 1.

If cosA=4/5 and sinA=-3/5 then sinA + 2cosA does equal 1 but 2sinA-cosA= -2.

So my conclusion is that if sinA+2cosA=1 then 2sinA-cosA can equal either 2 or -2.