Verify: -(cotA+cotB)/(cotA-cotB) = sin(A+B)/sin(A-B)cotA+cotBcotAcotB=sin(A+B)sin(AB) ?

2 Answers
Jul 30, 2016

Expand and simplify RHS then apply identities for sin(A+-B)sin(A±B)

Explanation:

RHS =-(cotA+cotB)/(cotA-cotB) = -(cosA/sinA+cosB/sinB)/(cosA/sinA-cosB/sinB)cotA+cotBcotAcotB=cosAsinA+cosBsinBcosAsinAcosBsinB

RHS=-(cosAsinB + cosBsinA)/(sinAsinB)/(cosAsinB-cosBsinA)/(sinAsinB)cosAsinB+cosBsinAsinAsinB/cosAsinBcosBsinAsinAsinB

RHS=-(cosAsinB + cosBsinA)/(cancel(sinAsinB))/(cosAsinB-cosBsinA)/(cancel(sinAsinB))

RHS=-(cosAsinB + cosBsinA)/(cosAsinB-cosBsinA)

=-sin(A+B)/-sin(A-B) = sin(A+B)/sin(A-B) = LHS

Jul 30, 2016

LHS=sin(A+B)/sin(A-B)

=(sinAcosB+cosAsinB)/(sinAcosB-cosAsinB)

Dividing both numerator and denominator by sinAsinB

=((sinAcosB)/(SinAsinB)+(cosAsinB)/(sinAsinB))/((sinAcosB)/(sinAsinB)-(cosAsinB)/(sinAsinB))

=-((cotA+cotB)/(cotA-cotB))=RHS

Verified