Evaluate the trig expression

Question

Evaluate:

(1) \(\dfrac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\tan 225^{\circ} \cdot \cos 200^{\circ}}\)

 

(2) \(\dfrac{\sin 36^{\circ}}{\sin 12^{\circ}}-\dfrac{\cos 36^{\circ}}{\cos 12^{\circ}}\)

Answer 1

(1)
\[
\begin{array}{ll}
\frac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\tan \left(180^{\circ}+45^{\circ}\right) \cdot \cos \left(180^{\circ}+20^{\circ}\right)} & \text { (reduce angles to acute angles ) } \\
=\frac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\tan 45^{\circ} \cdot\left(-\cos 20^{\circ}\right)} & \left(\cos 20^{\circ}=\cos \left(90^{\circ}-70^{\circ}\right)=\sin 70^{\circ}\right) \\
=\frac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{(1) \cdot\left(-\sin 70^{\circ}\right)} & \left(70^{\circ} \text { is double } 35^{\circ}\right) \\
=\frac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\left(-\sin \left[2\left(35^{\circ}\right)\right]\right)} & \text { (apply the double angle formula) } \\
=\frac{\sin 35^{\circ} \cdot \cos 35^{\circ}}{\left(-2 \sin 35^{\circ} \cdot \cos 35^{\circ}\right)} & \\
=-\frac{1}{2} &
\end{array}
\]

 

(2)
\[
\begin{aligned}
& \frac{\sin 36^{\circ} \cdot \cos 12^{\circ}}{\sin 12^{\circ} \cdot \cos 12^{\circ}}-\frac{\cos 36^{\circ} \cdot \sin 12^{\circ}}{\sin 12^{\circ} \cdot \cos 12^{\circ}} \quad\left(\mathrm{LCD}: \sin 12^{\circ} \cdot \cos 12^{\circ}\right) \\
& =\frac{\sin 36^{\circ} \cdot \cos 12^{\circ}-\cos 36^{\circ} \cdot \sin 12^{\circ}}{\sin 12^{\circ} \cdot \cos 12^{\circ}} \\
& =\frac{\sin \left(36^{\circ}-12^{\circ}\right)}{\sin 12^{\circ} \cdot \cos 12^{\circ}} \\
& =\frac{\sin 24^{\circ}}{\sin 12^{\circ} \cdot \cos 12^{\circ}} \\
& =\frac{\sin \left[2\left(12^{\circ}\right)\right]}{\sin 12^{\circ} \cdot \cos 12^{\circ}} \\
& =\frac{2 \sin 12^{\circ} \cdot \cos 12^{\circ}}{\sin 12^{\circ} \cdot \cos 12^{\circ}}=2
\end{aligned}
\]