Show work to algebraically find the exact value of all the solutions in the interval [0,2p) to tan(x + p) - cos(x + (p/2)) = 0

Question

Please Help! Find all solutions.

Answer 1

tan(x + π) = (tanx + tanπ)/[1-(tanx)(tanπ)] = tanx, since tanπ=0

 

cos(x + π/2)=cosx cos(π/2)-sinxsin(π/2)=-sinx, since

         cos(π/2) = 0 and sin(π/2) = 1 

 

So, the original equation simplifies to tanx + sinx = 0

 

sinx/cosx + sinx = 0

 

(sinx + cosxsinx)/cosx = 0

 

sinx + cosxsinx = 0

 

sinx(1 + cosx) = 0

 

sinx = 0 or cosx = -1     x = 0, π