Question

Evaluate $\frac{\sin {36}^o}{\cos {54}^o}-\frac{\sin {54}^o}{\cos {36}^o}$

Answer 1

Consider $\sin {36}^0$ and $\sin {54}^0$ which can be rewritten as follows:

$\sin {36}^0=\sin {({90}^0-54}^0)=\cos {54}^0$

$\sin {54}^0=\sin {({90}^0-36}^0)=\cos {36}^0$

$(\because \sin ({90}^0-x)=\cos x)$

Therefore, $\frac{\sin {36}^0}{\cos {54}^0}-\frac{\sin {54}^0}{\cos {36}^0}$ can be evaluated as shown below:

$\frac{\sin {36}^0}{\cos {54}^0}-\frac{\sin {54}^0}{\cos {36}^0}=\frac{\cos {54}^0}{\cos {54}^0}-\frac{\cos {36}^0}{\cos {36}^0}=1-1=0$

Hence, $\frac{\sin {36}^0}{\cos {54}^0}-\frac{\sin {54}^0}{\cos {36}^0}=0$.