Question

(sin²25° + sin²65°) + $\sqrt{3}$ (tan 5° tan 15° tan 30° tan 75° tan 85°)

Answer 1

$$\begin{gathered} \begin{array}{l}({\sin }^2{25}^0+{\sin }^2{65}^0)+\sqrt{3}\left(\tan 5^0\tan {15}^0\tan {30}^0\tan {75}^0\tan {85}^0\right)\\ \begin{array}{l}{=}^2{25}^0+{\sin }^2({90}^0-{25}^0)+\sqrt{3}\left(\tan {30}^0\right)\left(\tan 5^0\tan {85}^0\right)\left(\tan {15}^0\tan {75}^0\right)\\ \end{array}\end{array} \\ \begin{array}{l}=({\sin }^2{25}^0+{\cos }^2{25}^0)+\sqrt{3}\times \frac{1}{\sqrt{3}}\left\{\tan 5^0\tan ({90}^0-5^0)\right\}\left\{\tan {15}^0\tan ({90}^0-{15}^0)\right\}\\ \end{array} \\ \begin{array}{l}=1+1\left(\tan 5^0\cot 5^0\right)\left(\tan {15}^0\cot {15}^0\right)\\ \end{array} \\ \begin{array}{l}{\mathrm{=1+tan5}}^0\frac{1}{\tan 5^0}\tan {15}^0\frac{1}{\tan {15}^0}\\ \end{array} \\ \begin{array}{l}\mathrm{=1+1}\\ \end{array} \\ \mathrm{=2} \end{gathered}$$