Question

Without using trigonometrical tables, evaluate:

${\displaystyle \cos e{c}^2{57}^{\circ }-{\tan }^2{33}^{\circ }+{\cos 44}^{\circ }\cos ec{46}^{\circ }-\sqrt{2}{\cos 45}^{\circ }-{\tan }^2{60}^{\circ }}$

Answer 1

${\displaystyle \cos e{c}^2{57}^{\circ }-{\tan }^2{33}^{\circ }+{\cos 44}^{\circ }\cos ec{46}^{\circ }-\sqrt{2}{\cos 45}^{\circ }-{\tan }^2{60}^{\circ }}$

= ${\displaystyle \cos e{c}^2({90}^{\circ }-{33}^{\circ })-{\tan }^2{33}^{\circ }+{\cos 44}^{\circ }\cos ec({90}^{\circ }-{44}^{\circ })-\sqrt{2}{\cos 45}^{\circ }-{\tan }^2{60}^{\circ }}$

= ${\displaystyle {\sec }^2{33}^{\circ }-{\tan }^2{33}^{\circ }+{\cos 44}^{\circ }{\sec 44}^{\circ }-\sqrt{2}{\cos 45}^{\circ }-{\tan }^{2}60}$

= ${\displaystyle 1+1-\sqrt{2}{\cos 45}^{\circ }- {\tan }^{2}60}$

= ${\displaystyle 1+1-\sqrt{2}\left(\frac{1}{\sqrt{2}}\right)-{\left(\sqrt{3}\right)}^2}$

= 2 – 1 – 3

= – 2