Question

Prove : $\frac{3\cos {55}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\cos {70}^{\circ }.\cos ec{20}^{\circ }}{7\tan 5^{\circ }.\tan {25}^{\circ }.\tan {45}^{\circ }.\tan {65}^{\circ }.\tan {85}^{\circ }}=\frac{-1}{7}$

Answer 1

$\frac{3\cos {55}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\cos {70}^{\circ }.\cos ec{20}^{\circ }}{7\tan 5^{\circ }.\tan {25}^{\circ }.\tan {45}^{\circ }.\tan {65}^{\circ }.\tan {85}^{\circ }}$

$=\frac{3\cos \left({90}^{\circ }-{55}^{\circ }\right)}{7\sin {35}^{\circ }}-\frac{4\cos {70}^{\circ }\cos ec\left({90}^{\circ }-{70}^{\circ }\right)}{7\tan \left({90}^{\circ }-{85}^{\circ }\right)\tan \left({90}^{\circ }-{65}^{\circ }\right)\times 1\times \tan {65}^{\circ }\tan {85}^{\circ }}$

$=\frac{3\sin {35}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\cos {70}^{\circ }\times \sec {70}^{\circ }}{7\cot {85}^{\circ }\cot {65}^{\circ }\tan {65}^{\circ }\tan {85}^{\circ }}$

$=\frac{3}{7}-\frac{4\times \cos {70}^{\circ }\times \frac{1}{\cos {70}^{\circ }}}{7\times \frac{1}{\tan {85}^{\circ }\times \tan {65}^{\circ }}\times \tan {65}^{\circ }\times \tan {85}^{\circ }}$

$=\frac{3}{7}-\frac{4}{7}$

$=\frac{-1}{7}=R.H.S$

LHS=RHS

Hence proved.