Question

Evaluate -
1/cos² 60° + 1/sin² 60° - 1/2 cot² 45° -sin³ 90°

Answer 1

$$\begin{gathered} \frac{1}{{\cos }^2{60}^0}+\frac{1}{{\sin }^2{60}^0}-\frac{1}{2{\cot }^2{45}^0}-{\sin }^2{90}^0=\frac{1}{{\left({\displaystyle \frac{1}{2}}\right)}^2}+\frac{1}{{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2}-\frac{1}{2{\left({\displaystyle 1}\right)}^2}-{\left(1\right)}^2 \\ =4+\frac{4}{3}-\frac{1}{2}-1 \\ =\frac{24+8-3-6}{6} \\ =\frac{23}{6} \\ \Rightarrow \frac{\mathbf{1}}{{\mathbf{cos}}^{\mathbf{2}}{\mathbf{60}}^{\mathbf{0}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{sin}}^{\mathbf{2}}{\mathbf{60}}^{\mathbf{0}}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}{\mathbf{cot}}^{\mathbf{2}}{\mathbf{45}}^{\mathbf{0}}}\mathbf{-}{\mathbf{sin}}^{\mathbf{2}}{\mathbf{90}}^{\mathbf{0}}\mathbf{=}\frac{\mathbf{23}}{\mathbf{6}} \end{gathered}$$