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1 Lecture NotesTrigonometric identities 1page 1 Sample ProblemsProve each of the following + cosx= + tanx= sinxcos2x= 1 + sin +1 + sin cos = 2 sec sinx cosx1 + sinx= 2 + cos4xsin2x cos2x= + 1= sinxcosx=cosx1 + 2 cos2x=tan2x 1tan2x+ = csc2 tan2 + tanx=cosx1 sin cot tan = cos4x= 1 2 cos2x15.(sinx cosx)2+ (sinx+ cosx)2= + 4 sinx+ 3cos2x=3 + sinx1 sinx tanx= + 1 + tanxsecx=1 + sinxcos2xc copyright Hidegkuti, Powell, 2009 Last revised: May 8, 2013 Lecture NotesTrigonometric identities 1page 2 Practice ProblemsProve each of the following +cosx1 + sinx= + 1 = sinx 11 + sinx= 2 + cotx= + tan2x1 tan2x=1cos2x sin2x= cosxsinx+sinx1 cosx= 2 1secx+ 1=1 cosx1 + + cot2x= 1csc2x= 1cotx+ 1=1 tanx1 + tanx12.

2 (sinx+ cosx) (tanx+ cotx) = secx+ + cos3xsinx+ cosx= 1 + 1sin3x=cscx1 + sinx1 sinx 1 sinx1 + sinx= 4 cot4x= csc2x+ + 3 cosx+ 2=1 cosx2 + + tanycotx+ coty= + tanx1 tanx=cosx+ sinxcosx sinx20.(sinx tanx) (cosx cotx) = (sinx 1) (cosx 1)c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 2013 Lecture NotesTrigonometric identities 1page 3 Sample Problems - + cosx= secxSolution: We will only use the fact thatsin2x+ cos2x= 1for all values tanxsinx+ cosx=sinxcosx sinx+ cosx=sin2xcosx+ cosx=sin2xcosx+cos2xcosx=sin2x+ cos2xcosx=1cosx= + tanx=1sinxcosxSolution: We will only use the fact thatsin2x+ cos2x= 1for all values + tanx=cosxsinx+sinxcosx=cos2x+ sin2xsinxcosx=1sinxcosx= sinxcos2x= sin3xSolution.

3 We will only use the fact thatsin2x+ cos2x= 1for all values sinx sinxcos2x= sinx 1 cos2x = sinx sin2x= 1 + sin +1 + sin cos = 2 sec Solution: We will only use the fact thatsin2x+ cos2x= 1for all values 1 + sin +1 + sin cos =cos2 (1 + sin ) cos +(1 + sin )2(1 + sin ) cos =cos2 + (1 + sin )2(1 + sin ) cos =cos2 + 1 + 2 sin + sin2 (1 + sin ) cos =cos2 + sin2 + 1 + 2 sin (1 + sin ) cos =2 + 2 sin (1 + sin ) cos =2 (1 + sin )(1 + sin ) cos =2cos = 2 1cos = 2 sec = sinx cosx1 + sinx= 2 tanxSolution: We will start with the left-hand side. First we bring the fractions to the commondenominator.

4 Recall thatsin2x+ cos2x= 1for all values sinx cosx1 + sinx=cosx(1 + sinx)(1 sinx) (1 + sinx) cosx(1 sinx)(1 sinx) (1 + sinx)=cosx(1 + sinx) cosx(1 sinx)(1 sinx) (1 + sinx)=cosx+ cosxsinx cosx+ cosxsinx1 sin2x=2 sinxcosxcos2x=2 sinxcosx= 2 tanx=RHSc copyright Hidegkuti, Powell, 2009 Last revised: May 8, 2013 Lecture NotesTrigonometric identities 1page + cotxSolution: We will start with the right-hand side. We will re-write everything in terms ofsinxandcosxand simplify. We will again run into the Pythagorean identity,sin2x+ cos2x= + cotx=1sinx cosxsinxcosx+cosxsinx=1sinx cosx1sin2xsinxcosx+cos2xsinxcosx=cosxsin xsin2x+ cos2xsinxcosx=cosxsinx1sinxcosx=cosxsinx cosxsinx1=cos2x1= cos2x= cos4xsin2x cos2x= 1 Solution: We can factor the numerator via the di erence of squares cos4xsin2x cos2x= sin2x 2 (cos2x)2sin2x cos2x= sin2x+ cos2x sin2x cos2x sin2x cos2x= sin2x+ cos2x= 1 = + 1= sin2xSolution.

5 LHS=tan2xtan2x+ 1= sinxcosx 2 sinxcosx 2+ 1=sin2xcos2xsin2xcos2x+ 1=sin2xcos2xsin2xcos2x+cos2xcos2x=sin2xc os2xsin2x+ cos2xcos2x=sin2xcos2x1cos2x=sin2xcos2x cos2x1= sin2x= sinxcosx=cosx1 + sinxSolution:LHS=1 sinxcosx=1 sinxcosx 1 =1 sinxcosx 1 + sinx1 + sinx=(1 sinx) (1 + sinx)cosx(1 + sinx)=1 sin2xcosx(1 + sinx)=cos2xcosx(1 + sinx)=cosx1 + sinx=RHSc copyright Hidegkuti, Powell, 2009 Last revised: May 8, 2013 Lecture NotesTrigonometric identities 1page 2 cos2x=tan2x 1tan2x+ 1 Solution:RHS=tan2x 1tan2x+ 1=sin2xcos2x 1sin2xcos2x+ 1=sin2xcos2x cos2xcos2xsin2xcos2x+cos2xcos2x=sin2x cos2xcos2xsin2x+ cos2xcos2x=sin2x cos2xcos2x cos2xsin2x+ cos2x=sin2x cos2xsin2x+ cos2x=sin2x cos2x1= sin2x cos2x= 1 cos2x cos2x= 1 2 cos2x= = csc2 tan2 1 RHS= csc2 tan2 1 =1sin2 sin cos 2 1 =1sin2 sin2 cos2 1 =1cos2 1=1cos2 cos2 cos2 =1 cos2 cos2 =sin2 cos2 = sin cos 2= tan2 = + tanx=cosx1 sinxSolution.

6 RHS=cosx1 sinx=cosx1 sinx 1 =cosx1 sinx 1 + sinx1 + sinx=cosx(1 + sinx)(1 sinx) (1 + sinx)=cosx(1 + sinx)1 sin2x=cosx(1 + sinx)cos2x=1 + sinxcosx=1cosx+sinxcosx= sin cot tan = 1 Solution: We will start with the left-hand side. We will re-write everything in terms ofsin andcos and simplify. We will again run into the Pythagorean identity,sin2x+ cos2x= 1for all sin cot tan =1sin sin 1 cos sin sin cos =1sin 1sin cos sin cos sin =1sin2 cos2 sin2 =1 cos2 sin2 = sin2 + cos2 cos2 sin2 =sin2 sin2 = 1 = cos4x= 1 2 cos2xSolution:LHS= sin4x cos4x= sin2x 2 cos2x 2= sin2x+ cos2x sin2x cos2x = 1 sin2x cos2x = 1 cos2x cos2x= 1 2 cos2x=RHSc copyright Hidegkuti, Powell, 2009 Last revised: May 8, 2013 Lecture NotesTrigonometric identities 1page 615.

7 (sinx cosx)2+ (sinx+ cosx)2= 2 Solution:LHS= (sinx cosx)2+ (sinx+ cosx)2= sin2x+ cos2x 2 sinxcosx + sin2x+ cos2x+ 2 sinxcosx = 2 sin2x+ 2 cos2x= 2 sin2x+ cos2x = 2 1 = 2 = + 4 sinx+ 3cos2x=3 + sinx1 sinxSolution:LHS=sin2x+ 4 sinx+ 3cos2x=(sinx+ 1) (sinx+ 3)1 sin2x=(sinx+ 1) (sinx+ 3)(1 + sinx) (1 sinx)=sinx+ 31 sinx= sinx tanx= secxSolution:LHS=cosx1 sinx tanx=cosx1 sinx sinxcosx=cos2x sinx(1 sinx)cosx(1 sinx)=cos2x sinx+ sin2xcosx(1 sinx)= cos2x+ sin2x sinxcosx(1 sinx)=1 sinxcosx(1 sinx)=1cosx= + 1 + tanxsecx=1 + sinxcos2xSolution:LHS= tan2x+ 1 + tanxsecx=sin2xcos2x+ 1 +sinxcosx 1cosx=sin2xcos2x+cos2xcos2x+sinxcos2x=si n2x+ cos2x+ sinxcos2x=1 + sinxcos2x=RHSFor more documents like this, visit our page at and click onLecture Notes.

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