sin+2cos=1 ,, then prove thatcos-2sin=2

1.	sin+2cos=1 ,, then prove thatcos-2sin=2
Answer» Given that Sin A + 2 cos A = 1 Squaring on both sides, we get (sin A + 2 cos A)^2 = 1 We know that (a+b)^2 = a^2 + b^2 + 2ab. (sin^2 A + 4 cos^2 A + 4 sin A cos A) = 1 4 cos^2 A + 4 sin A cos A = 1 - sin^2 A 4 cos^2 A + 4 sin A cos A = cos^2 A 3 cos^2 A + 4 sin A cos A = 0 3 cos^2 A = - 4 sin A cos A ---- (1). Given 2 sin A - cos A Squaring on both sides, we get (2 sin A - cos A)^2 = 4 sin^2 A + cos^2 A - 4 sin A cos A = 4 sin^2 A + cos^2 A + 3 cos^2 A = 4 sin^2 A + 4 cos^2 A = 4(sin^2 A + cos^2 A) = 4. 2 sin A - cos A = 2.LHS = RHS.

Answer»

Given that Sin A + 2 cos A = 1

Squaring on both sides, we get

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

We know that (a+b)^2 = a^2 + b^2 + 2ab.

(sin^2 A + 4 cos^2 A + 4 sin A cos A) = 1

4 cos^2 A + 4 sin A cos A = 1 - sin^2 A

4 cos^2 A + 4 sin A cos A = cos^2 A

3 cos^2 A + 4 sin A cos A = 0

3 cos^2 A = - 4 sin A cos A ---- (1).

Given 2 sin A - cos A

Squaring on both sides, we get

(2 sin A - cos A)^2 = 4 sin^2 A + cos^2 A - 4 sin A cos A

= 4 sin^2 A + cos^2 A + 3 cos^2 A

= 4 sin^2 A + 4 cos^2 A

= 4(sin^2 A + cos^2 A)

= 4.

2 sin A - cos A = 2.LHS = RHS.

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