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If sinA +2cosA=1 prove that 2sinA-cosA=2

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If sinA +2cosA=1 prove that 2sinA-cosA=2
  • 1 answers

Yogita Ingle 5 years, 3 months ago

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 = a2 + b2 + 2ab.

(sin2 A + 4 cos2 A + 4 sin A cos A) = 1

4 cos2 A + 4 sin A cos A = 1 - sin2 A

4 cos2 A + 4 sin A cos A = cos2 A

3 cos2 A + 4 sin A cos A = 0    

3 cos2 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 sin2 A + cos2 A - 4 sin A cos A

                               = 4 sin2 A + cos2 A + 3 cos2 A

                               = 4 sin2 A + 4 cos2 A

                               = 4(sin2 A + cos2 A)

                               = 4.

2 sin A - cos A = 2.
 

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