Question

Evaluate: $co{t}^2{30}^0-2{\cos }^2{60}^0-\frac{3}{4}se{c}^2{45}^0-4se{c}^2{30}^0$

Answer 1

Evaluating the given expression:

$$\begin{gathered} co{t}^2{30}^0-2{\cos }^2{60}^0-\frac{3}{4}se{c}^2{45}^0-4se{c}^2{30}^0 \\ ={\left(\sqrt{3}\right)}^2-2{\left(\frac{1}{2}\right)}^2-\frac{3}{4}{\left(\sqrt{2}\right)}^2-4\left(\frac{2}{\sqrt{3}}\right)\left[cot30°=\sqrt{3},\cos 60°=\frac{1}{2},sec45°=\sqrt{2},sec30°=\frac{2}{\sqrt{3}}\right] \\ =3-\frac{1}{2}-\frac{3}{2}-\frac{8}{\sqrt{3}} \\ =\frac{6-1-3}{2}-\frac{8}{\sqrt{3}} \\ =1-\frac{8}{\sqrt{3}} \\ =\frac{\left(\sqrt{3}-8\right)}{\sqrt{3}} \end{gathered}$$