Prove/verify the identity? #sin(alpha-beta)/(cos(alpha+beta)) = (tanalpha-tanbeta)/(1-tanalphatanbeta)#

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1 Answer
Hriman
Apr 7, 2018

verified below...

Explanation:

#sin(a-b)/cos(a+b)= (tana-tanb)/(1-tanatanb)#

Apply sine difference and cosine sum identities:
#(sinacosb-cosasinb)/(cosacosb-sinasinb)= (tana-tanb)/(1-tanatanb)#

Divide each term in the numerator by #cosacosb#:
#(cosacosb((sinacancel(cosb))/(cosacancel(cosb))-(cancel(cosa)sinb)/(cancel(cosa)cosb)))/(cosacosb-sinasinb)= (tana-tanb)/(1-tanatanb)#

Apply quotient identity: #sintheta/costheta=tantheta#
#(cosacosb(tana-tanb))/(cosacosb-sinasinb)= (tana-tanb)/(1-tanatanb)#

Divide each term in the denominator by #cosacosb#:
#(cosacosb(tana-tanb))/(cosacosb((cancel(cosacosb))/(cancel(cosacosb))-(sinasinb)/(cosacosb)))= (tana-tanb)/(1-tanatanb)#

Apply quotient identity: #sintheta/costheta=tantheta#:
#(cancel(cosacosb)(tana-tanb))/(cancel(cosacosb)(1-tanatanb))= (tana-tanb)/(1-tanatanb)#

#(tana-tanb)/(1-tanatanb)= (tana-tanb)/(1-tanatanb)#

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