Question

Cos²20⁰+cos²70⁰ + 2 cosec²58⁰ - 2 cot 58⁰ tan 32⁰ – 4 tan 13⁰ tan 37⁰ tan 45⁰ tan 53⁰ tan 77⁰
Sec²50⁰ - cot² 40⁰

Answer 1

$$\begin{gathered} \frac{{\cos }^{2}20+{\cos }^{2}70}{{\sec }^{2}50-{\cot }^{2}40}+2{\mathrm{cosec}}^{2}58-2\cot 58.\tan 32-4\tan 13.\tan 37.\tan 45.\tan 53.\tan 77 \\ =\frac{{\cos }^2\left(90-70\right)+{\cos }^{2}70}{{\sec }^{2}50-{\cot }^2\left(90-50\right)}+2{\mathrm{cosec}}^{2}58-2\cot 58.\tan \left(90-58\right)-4\tan \left(90-77\right).\tan \left(90-53\right).1.\tan 53.\tan 77 \\ =\frac{{\sin }^{2}70+{\cos }^{2}70}{{\sec }^{2}50-{\tan }^{2}50}+2{\mathrm{cosec}}^{2}58-2\cot 58.\cot 58-4\cot 77.\cot 53.\tan 53.\tan 77 \\ =\frac{1}{1}+2\left[{\mathrm{cosec}}^{2}58-{\cot }^{2}58\right]-4\times \frac{1}{\tan 77}\times \frac{1}{\tan 53}\times \tan 53\times \tan 77 \\ =1+2-4 \\ =-1 \end{gathered}$$