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 the formula \[cot\left(A+B\right)=\frac{cotAcotB-1}{cotA+cotB}=>cotAcotB=cot\left(A+B\right)\left(cotA+\cot{B}\right)+1\] Substituting into the above expression \[\rightarrow \left\{cot\left(\frac{\pi}{20}+\frac{9\pi}{20}\right)\left(cot\frac{\pi}{20}+\cot{\frac{9\pi}{20}}\right)+1\right\}\…

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