Question

Evaluate the following:
$\frac{\sin {36}^{\circ }}{\cos {54}^{\circ }}-\frac{\sin {54}^{\circ }}{\cos {36}^{\circ }}$

Answer 1

We will use $\cos (90-\theta )=\sin \theta$ to solve this question.

$\begin{array}{cc}\frac{\sin {36}^{\circ }}{\cos {54}^{\circ }}-\frac{\sin {54}^{\circ }}{\cos {36}^{\circ }}& =\frac{\sin {36}^{\circ }}{\cos \left({90}^{\circ }-{36}^{\circ }\right)}-\frac{\sin {54}^{\circ }}{\cos \left({90}^{\circ }-{54}^{\circ }\right)}\\ & \\ & =\frac{\sin {36}^{\circ }}{\sin {36}^{\circ }}-\frac{\sin {54}^{\circ }}{\sin {54}^{\circ }}\\ & \\ & =1-1\\ & \\ & =0\end{array}$

Hence the given expression is equal to $0.$