Question
Write the value of $\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ?$
✓
Answer
We have to find: $\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ$
$=\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ$
$=\tan\big(90^\circ-80^\circ\big)\tan\big(90^\circ-75^\circ\big)\tan75^\circ\tan80^\circ$
$=\cot80^\circ\cot75^\circ\tan75^\circ\tan80^\circ$
$=\big(\cot75^\circ\tan75^\circ\big)\big(\cot80^\circ\tan80^\circ\big)$ $\big[\tan\big(90^\circ-\theta\big)=\cot\theta\big]$
$=1\times1$
$=1$ $\big[\cot\theta.\tan\theta=1\big]$
Hence the value of $\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ\text{ is }1$
$=\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ$
$=\tan\big(90^\circ-80^\circ\big)\tan\big(90^\circ-75^\circ\big)\tan75^\circ\tan80^\circ$
$=\cot80^\circ\cot75^\circ\tan75^\circ\tan80^\circ$
$=\big(\cot75^\circ\tan75^\circ\big)\big(\cot80^\circ\tan80^\circ\big)$ $\big[\tan\big(90^\circ-\theta\big)=\cot\theta\big]$
$=1\times1$
$=1$ $\big[\cot\theta.\tan\theta=1\big]$
Hence the value of $\tan10^\circ\tan15^\circ\tan75^\circ\tan80^\circ\text{ is }1$
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