find the general solution of equation sinx-3sin2x+sin3x=cos x-3 cos 2X+cos3x - Maths - Trigonometric Functions

Question

find the general solution of equation sinx-3sin2x+sin3x=cos x-3
cos 2X+cos3x

Prasanta · Accepted Answer

sinx-3sin2x+sin3x=cosx-3cos2x+cos3x

⇒sinx+sin3x-3sin2x-cosx+cos3x+3cos2x=0

⇒2sinx+3x2cosx-3x2-3sin2x-2cosx+3x2cosx-3x2+3cos2x=0

⇒2sin2xcosx-3sin2x-2cos2xcosx+3cos2x=0

⇒sin2x2cosx-3-cos2x2cosx-3=0

⇒2cosx-3sin2x-cos2x=0

⇒cosx=32, or, sin2x=cos2x

As, cos x∈[-1,1] Thus,cos x≠32

So,

sin2x=cos2x⇒tan2x=1⇒tan2x=tanπ4

⇒2x=nπ+π4

⇒x=nπ2+π8