SOLUTION: If tan(x/2)=sinx and cosx≠0, then the value of tanx is…

One obvious solution is 





Divide both sides by , assuming , where
 is true in the obvious solution above.





















, where n is any integer.

and also the obvious solution 

Edwin

Our starting equation is 

     = .      (1)


Since  is not defined when  = 0, the domain of this equation is 
the set of all real numbers except  = ,  or  x =  = .


One obvious solution to equation (1) is x = , when both sides of equation (1) are zeroes.    (2)


In what follows, we assume that x is neither  nor .    (3)


        In particular, it implies that  =/= 0.    (4)


Now we transform equation (1) step by step

     = 

     = 


Due to (4), we can divide both sides by , since this factor is not zero.

     = 

     = 

     = 

     =  = 

     =     <<<---=== notice here the difference with the Edwin's solution
                                 (here Edwin lost some/(half of the) solutions)


     =  =  = , where n is any integer.

              But for all these values of x, cos(x) = 0, so we should exclude this set of values
              from our consideration due to condition cos(x) =/= 0, imposed in the problem.


Finally, the only set of solutions is  x = , found in (2).


ANSWER.  The solution to equation (1) with the imposed condition cos(x) =/= 0 

         is the set  x = , and tan(x) = 0 for all these values of x.