CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Prove that:
(i) sin(A+B)+sin(AB)cos(A+B)+cos(AB)=tanA
(ii) sin(AB)cosAcosB+sin(BC)cosBcosC+sin(CA)cosCcosA=0
(iii) sin(AB)sinAsinB+sin(BC)sinBsinC+sin(CA)sinCsinA=0
(iv) sin2B=sin2A+sin2(AB)2sinAcosBsin(AB)
(v) cos2+cos2B2cosAcosBcos(A+B)=sin2(A+B)
(vi) tan(A+B)cot(AB)=tan2Atan2B1tan2Atan2B


Solution

(i) We have,
LHS:sin(A+B)+sin(AB)cos(A+B)+cos(AB)
=2sinAcosB2cosAcosB
=sinAcosA
=tanA
=RHS
LHS=RHS
Hence proved.
(ii) We have,
LHS:sin(AB)cosAcosB+sin(BC)cosBcosC+sin(CA)cosCcosA
=sinAcosBcosAsinBcosAcosB+sinBcosCcosBsinCcosBcosC+sinCcosAcosCsinAcosCcosA
=sinAcosBcosAcosBcosAsinBcosAcosB+sinBcosCcosBcosCsinBsinCcosBcosC+sinCcosAcosCcosAcosCsinAcosCcosA
=tanA-tanB+tanB-tanC+tanC-tanA
=0
=RHS
LHS=RHS
Hence proved.
(iii)We have,
LHS= sin(AB)sinAsinB+sin(BC)sinBsinC+sin(CA)sinCsinA
sinAcosBcosAsinBsinAsinB+sinBsinCcosBsinCsinBsinB+sinCcosAcosCsinAsinCsinA
=sinAcosBsinAsinBcosAsinBsinAsinB+sinBcosCsinBsinCcosBsinCsinBsinC+sinCcosAsinCsinAcosCsinAsinCsinA
=cosBsinBcosAsinA+cosCsinCcosBsinB+cosAsinAcosCsinC
=cotB-cotA+cotC-cotB+cotA-cotC
=0
=RHS
LHS=RHS
Hence proved. 
(iv)We have,
RHS:sin2A+sin2(AB)2sinAcosBsin(AB)
=sin2A=sin(AB)[sin(AB)2sinAcosB]
=sin2A+sin(AB)[sinAcosBcosAsinB2sinAcosB]
=sin2A+sin(AB)[sinAcosBcosAsinB]
=sin2Asin(AB)(sinAcosB+cosAsinB)
=sin2Asin(AB)(sin(A+B))
=sin2Asin(AB)sin(A+B)
=sin2A(sin2Asin2B)
=sin2Asin2A+sin2B
=sin2B
=LHS
LHS=RHS
Hence proved.
(v)RHS =cos2A+cos2B2cosAcosBcos(A+B)
=cos2A+(1sin2B)2cosAcosBcos(A+B)
=[cos2Asin2B]2cosAcosBcos(A+B)+1
=[cos(A+B)cos(AB)]2cosAcosBcos(A+B)+1
=cos(A+B)[cos(AB)2cosAcosB]+1
=cos(A+B)[cosAcosBsinAsinB2cosAcosB]+1
=cos(A+B)[cosAcosB+sinAsinB]+1
=-cos(A+B)[cos(A+B)]+1
=-cos2(A+B)+1
=-cos2A+B
= 1cos2(A+B)
= sin2(A+B)
=RHS
LHS=RHS
Hence proved.
(vi) We have,
LHS= tan(A+B)cot(AB)
=tan(A+B)1tan(AB)
=tan(A+B)tan(AB)
=[tanA+tanB1tanAtanB][tanAtanB1+tanAtanB]
(tanA+tanB)(tanAtanB)(1tanAtanB)(1+tanAtanB)
=tan2Atan2B1(tanAtanB)2
=tan2Atan2B1tan2Atan2B
=RHS
LHS=RHS
Hence proved.

Mathematics
RD Sharma
Standard XI

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
Same exercise questions
View More


similar_icon
People also searched for
View More



footer-image