Does $A\sin b=B\sin b+C\cos b$ imply $C=0$?

Question

If $\cos b$ and $\sin b$ are both non-zero, does $$A\sin b=B\sin b+C\cos b \quad\implies\quad C=0$$ because there is no $\cos$ term in the l.h.s. of the equation?

Thank you!!

Answer 1

As $\sin b \ne0 $ and $\cos b \ne0 $, you can divide by $\sin b $ to get $$ A = B + C\cot b$$ Rearranging this gives $$ C = (A-B)\tan b $$ So $$A\sin b=B\sin b+C\cos b $$ does not imply $C=0$ unless $A=B$, as it is given $\sin b \ne0 \iff \tan b \ne 0$.