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 \[\tan{\frac{5\pi}{20}}\ =\tan{\frac{\pi}{4}}=1\] \[ tan\frac{\pi}{20}.tan\frac{3\pi}{20}.tan\frac{7\pi}{20}.tan\frac{9\pi}{20}\] \[tan\frac{\pi}{20}.tan\frac{9\pi}{20}.tan\frac{3\pi}{20}.tan\frac{7\pi}{20}\] \[\left(tan\frac{\pi}{20}.tan\frac{9\pi}{20}\right).\left(tan\frac{3\pi}{20}.tan\frac{7\pi}{20}\right)\] By the formula \[\tan{\left(A+B\right)}=\frac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}=>\tan{A}\tan{B}=\frac{\tan{A}+\tan{B}}{\tan{\left(A+B\right)}}+1\] Substituting into the above expression \[\left(\frac{tan\frac{\pi}{20}+\tan{\frac{9\pi}{20}}}{tan\left(\frac{\pi}{20}+\frac{9\pi}{20}\right)}+1\right)\…

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