Example 1
Prove that sin4x+cos4x=14(3+cos4x).(x+y)2=x2+y2+2xysin2x=1−cos2x2sin2x+cos2x=12sinxcosx=sin2x
Proof:
(sin2x+cos2x)2=sin4x+cos4x+2sin2xcos2x12=sin4x+cos4x+12(2sinxcosx)21=sin4x+cos4x+12sin22x=sin4x+cos4x+12×1−cos4x2sin4x+cos4x=1−(14−cos4x4)=34+cos4x4Hence sin4x+cos4x=14(3+cos4x).
Example 2
If sinx+cosx=a, then show that sin6x+cos6x=14(4−3(a2−1)2).(x+y)3=x3+y3+3xy(x+y)(x+y)2=x2+y2+2xy1=sin2x+cos2x
Proof:
a2=(sinx+cosx)2=sin2x+cos2x+2sinxcosx=1+2sinxcosxsinxcosx=a2−12(sin2x+cos2x)3=(sin2x)3+(cos2x)3+3sin2xcos2x(sin2x+cos2x)13=sin6x+cos6x+3(sinxcosx)2×11=sin6x+cos6x+3(a2−12)2sin6x+cos6x=1−34(a2−1)2=14[4−3(a2−1)2]
Example 3
If cosx−sinx=√2sinx, show that cosx+sinx=√2cosx.Proof:
cosx=sinx+√2sinx⋯(1)(1)×√2:√2cosx=√2sinx+2sinx⋯(2)(2)−(1):√2cosx−cosx=sinxHence √2cosx=sinx+cosx.
Example 4
Prove that cos3x+sin3xcosx−sinx=1+2sin2x.cosx−cosy=−2sinx+y2sinx−y2sinx+siny=2sinx+y2cosx−y2
Proof:
cos3x−cosx=−2sin3x+x2sin3x−x2=−2sin2xsinx⋯(1)sin3x+sinx=2sin3x+x2cos3x−x2=2sin2xcosx⋯(2)(1)+(2): cos3x+sin3x−(cosx−sinx)=2sin2x(cosx−sinx).
÷(cosx−sinx):cos3x+sin3xcosx−sinx−1=2sinx.
Therefore cos3x+sin3xcosx−sinx=1+2sinx.
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