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Trigonometry (Identity)

Example 1 

Prove that sin4x+cos4x=14(3+cos4x).

(x+y)2=x2+y2+2xysin2x=1cos2x2sin2x+cos2x=12sinxcosx=sin2x

Proof: 

(sin2x+cos2x)2=sin4x+cos4x+2sin2xcos2x12=sin4x+cos4x+12(2sinxcosx)21=sin4x+cos4x+12sin22x=sin4x+cos4x+12×1cos4x2sin4x+cos4x=1(14cos4x4)=34+cos4x4
Hence sin4x+cos4x=14(3+cos4x).

Example 2 

If sinx+cosx=a, then show that sin6x+cos6x=14(43(a21)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=a212
(sin2x+cos2x)3=(sin2x)3+(cos2x)3+3sin2xcos2x(sin2x+cos2x)13=sin6x+cos6x+3(sinxcosx)2×11=sin6x+cos6x+3(a212)2sin6x+cos6x=134(a21)2=14[43(a21)2]

Example 3 

If cosxsinx=2sinx, show that cosx+sinx=2cosx.

Proof: 

cosx=sinx+2sinx(1)(1)×2:2cosx=2sinx+2sinx(2)(2)(1):2cosxcosx=sinx
Hence 2cosx=sinx+cosx.

Example 4

Prove that cos3x+sin3xcosxsinx=1+2sin2x.
cosxcosy=2sinx+y2sinxy2sinx+siny=2sinx+y2cosxy2

Proof:

cos3xcosx=2sin3x+x2sin3xx2=2sin2xsinx(1)sin3x+sinx=2sin3x+x2cos3xx2=2sin2xcosx(2)

(1)+(2): cos3x+sin3x(cosxsinx)=2sin2x(cosxsinx).
÷(cosxsinx):cos3x+sin3xcosxsinx1=2sinx.
Therefore cos3x+sin3xcosxsinx=1+2sinx.

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