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Question

Find the general solution of (2 sin x - cos x) (1 + cos x) = sin2 x


A

+

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B

(2n + 1)

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C

2n

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D

+

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Solution

The correct options are
A

+


B

(2n + 1)


We can either expand L.H.S. and proceed or write sin2 x = 1 - cos2 x and proceed. We write

sin2 x = 1 - cos2x, because there is one (1 + cos x) on L.H.S and we can cancel a common factor. We will

proceed in this way because; we don't have to expand L.H.S.

(2 sin x - cos x)(1 + cos x) = 1 - cos2x

= (1 + cos x) (1 - cos x)

(One important point to note here is that when we cancel (1+ cos x) from both the sides, we should equate it to zero. Otherwise we will miss one solution)
2 sin x - cos x = 1 - cos x or 1+ cos x = 0
sin x = 12 or cos x = - 1
x=nπ + (1)n π6 or x = (2n + 1) π
Key concepts/6 steps:
(1) sin2 x = 1 - cos2 x

(2) Equating the common factor to zero

(3) sin x = sinα

x=nπ+(1)nαand
cos x = -1
x = (2n+1) π


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