Question

$\frac{\sin 15°\cos 75°+\cos 15°\sin 75°}{\tan 5°\tan 30°\tan 35°\tan 55°\tan 85°}$

Answer 1

$$\begin{gathered} \begin{array}{l}\frac{\sin {15}^0\cos {75}^0+\cos {15}^0\sin {75}^0}{\tan 5^0\tan {30}^0\tan {35}^0\tan {55}^0\tan {85}^0}\\ \end{array} \\ \begin{array}{l}=\frac{\sin {15}^0\cos ({90}^0-{15}^0)+\cos {15}^0\sin ({90}^0-{15}^0)}{\tan {30}^0\left\{\tan 5^0\tan {35}^0\tan {55}^0\tan {85}^0\right\}}\\ \end{array} \\ \begin{array}{l}=\frac{\sin {15}^0\sin {15}^0+\cos {15}^0\cos {15}^0}{\frac{1}{\sqrt{3}}\{\tan ({90}^0-5^0)\tan {85}^0.\tan {35}^0\tan ({90}^0-{35}^0)\}}\\ \end{array} \\ \begin{array}{l}\text{=}\frac{{\sin }^2{15}^0+{\cos }^2{15}^0}{\frac{1}{\sqrt{3}}\left(\cot {85}^0\tan {85}^0\right)(\tan {35}^0.\cot {35}^0)}\\ \end{array} \\ \begin{array}{l}\text{=}\frac{\sqrt{3}}{\left(\frac{1}{\tan {85}^0}\tan {85}^0\right)(\frac{1}{\cot {35}^0}.\cot {35}^0)}\\ \end{array} \\ \begin{array}{l}\text{=}\frac{\sqrt{3}}{1\times 1}\\ \\ \end{array} \\ \text{=}\sqrt{3} \end{gathered}$$