prove that tan pi/20.tan 3pi/20.tan 5pi/20.tan 7pi/20.tan 9pi/20=1

Question : Prove that tan pi/20.tan 3pi/20.tan 5pi/20.tan 7pi/20.tan 9pi/20=1 Solution : To prove that tan pi/20.tan 3pi/20.tan 5pi/20.tan 7pi/20.tan 9pi/20=1 LHS tan\frac{\pi}{20}.tan\frac{3\pi}{20}.\tan{\frac{5\pi}{20}}.tan\frac{7\pi}{20}.tan\frac{9\pi}{20} Since \tan{\frac{5\pi}{20}}\ =\tan{\frac{\pi}{4}}=1 tan\frac{\pi}{20}.tan\frac{3\pi}{20}.tan\frac{7\pi}{20}.tan\frac{9\pi}{20} tan\frac{\pi}{20}.tan\frac{9\pi}{20}.tan\frac{3\pi}{20}.tan\frac{7\pi}{20} \left(tan\frac{\pi}{20}.tan\frac{9\pi}{20}\right).\left(tan\frac{3\pi}{20}.tan\frac{7\pi}{20}\right)

solved prove that cot pi/20.cot 3pi/20.cot 5pi/20.cot 7pi/20.cot 9pi/20=1

Question : Prove that cot pi/20.cot 3pi/20.cot 5pi/20.cot 7pi/20.cot 9pi/20=1 Solution : To prove that cot pi/20.cot 3pi/20.cot 5pi/20.cot 7pi/20.cot 9pi/20=1 LHS cot\frac{\pi}{20}.cot\frac{3\pi}{20}.cot\frac{5\pi}{20}.cot\frac{7\pi}{20}.cot\frac{9\pi}{20} Since cot\frac{5\pi}{20}=\cot{\frac{\pi}{4}}=1 cot\frac{\pi}{20}.cot\frac{3\pi}{20}.cot\frac{7\pi}{20}.cot\frac{9\pi}{20} cot\frac{\pi}{20}.cot\frac{9\pi}{20}.cot\frac{3\pi}{20}.cot\frac{7\pi}{20} \rightarrow \left(cot\frac{\pi}{20}.cot\frac{9\pi}{20}\right).\left(cot\frac{3\pi}{20}.cot\frac{7\pi}{20}\right) By…