If $$sin$$⁡α$$=Asin$$⁡$$($$α$$+$$β$$)$$,A≠$$0$$, thenThe value of $$tan$$⁡α isThe value of $$tan$$⁡β isWhich of the following is not the value of $$tan$$⁡$$($$α$$+$$β$$)$$ ?

Question

If $$sin$$⁡α$$=Asin$$⁡$$($$α$$+$$β$$)$$,A≠$$0$$,  thenThe value of $$tan$$⁡α isThe value of $$tan$$⁡β isWhich of the following is not the value of $$tan$$⁡$$($$α$$+$$β$$)$$ ?

Infinity Learn · Accepted Answer

$$sin$$ α−Asin $$($$α$$+$$β$$)$$−$$A($$$$sin$$ α $$cos$$ β$$+$$$$sin$$ β $$cos$$ α$$)$$⇒$$sin$$α(1$$−Acosβ$$)=Asin$$β $$cos$$α.....$$(i)⇒$$tan$$⁡α$$=Asin$$⁡β(1$$−Acos⁡β$$).....(ii)$$tan$$⁡β$$=$$$$sin$$⁡β$$cos$$⁡β$$=(1$$−Acos⁡β$$)$$$$tan$$⁡αAcos⁡β$$=(1$$−Acos⁡β$$)$$$$sin$$⁡αAcos⁡α$$cos$$⁡β  [from Eqs. (i)and(ii)] $$tan$$⁡$$($$α$$+$$β$$)=$$$$tan$$⁡α$$+$$$$tan$$⁡β$$1$$−$$tan$$⁡α$$tan$$⁡β$$=Asin$$⁡β$$1$$−Acos⁡β$$+$$$$sin$$⁡β$$cos$$⁡β$$1$$−Asin⁡β$$sin$$⁡β(1$$−Acos⁡β$$)$$cos$$⁡β$$=Asin$$⁡β$$cos$$⁡β$$+$$$$sin$$⁡β−Asin⁡β$$cos$$⁡β$$cos$$⁡β−$$Acos2$$⁡β−$$Asin2$$⁡β$$=$$$$sin$$⁡β$$cos$$⁡β−AAlso, $$tan$$⁡$$($$α$$+$$β$$)=$$$$tan$$⁡α$$+$$$$tan$$⁡β$$1$$−$$tan$$⁡α$$tan$$⁡β$$=$$$$sin$$⁡α$$cos$$⁡α$$+$$$$sin$$⁡α(1$$−Acos⁡β$$)Acos$$⁡α$$cos$$⁡β$$1$$−$$sin2$$⁡α$$(1$$−Acos⁡β$$)Acos2$$⁡α$$cos$$⁡β[from Eq. $$(ii)]$$=$$[Asin⁡α$$cos$$⁡β$$+$$$$sin$$⁡α−Asin⁡α$$cos$$⁡β]$$cos$$⁡α$$Acos2$$⁡α$$cos$$⁡β−$$sin2$$⁡α$$+Asin2$$⁡α$$cos$$⁡β$$=$$$$sin$$⁡α$$cos$$⁡αAcos⁡β−$$sin2$$⁡α

If sin⁡α=Asin⁡(α+β),A≠0, thenThe value of tan⁡α isThe value of tan⁡β isWhich of the following is not the value of tan⁡(α+β) ?

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