Find the general solutions for 3tanx+cotx=5cosecxFind the general solution of tanax=cotbx

 Steps to solve the general solutions for 3tanx+cotx=5cosecx

⇒3tanx+cotx=5cosecx

⇒3(sinx/cosx)+(cosx/sinx)=5/sinx

⇒(3sin^x+cos^2x)/sinx.cosx=5/sinx

⇒3sin^2x+cos^2x=5cosx

⇒3(1-cos^2x)+cos^x=5cosx

⇒3-5cosx-2cos^2x=0

⇒3-6cosx+cosx-2cos^2x=0

⇒3(1-2cosx)+cosx(1-2cosx)=0

⇒(1-2cosx)(3+cosx)=0

⇒1-cosx=0 or ⇒3+cosx=0

⇒x=2nπ±π/3

Hence the general solution for 3tanx+cotx=5cosecx is 2nπ±π/3

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