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What is the value of $\tan {7^ \circ }\tan {23^ \circ }\tan {60^ \circ }\tan {67^ \circ }\tan {83^ \circ }$?
$
  a)\,\dfrac{1}{{\sqrt 3 }} \\
  b)\,\sqrt 3 \\
  c)\,1 \\
  d)\,\infty \\
 $

seo-qna
Last updated date: 20th Jun 2024
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Answer
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Hint:You should know that $\tan \left( {90 - \theta } \right) = \cot \theta \,\,\,\& \,\,\tan \theta = \dfrac{1}{{\cot \theta }}\,$or $\tan \theta . \cot \theta = 1$ using these formulas you can get the required answer.

Formula used:
Complete step-by-step answer:
According to the question we need to find the value of $\tan {7^ \circ }\tan {23^ \circ }\tan {60^ \circ }\tan {67^ \circ }\tan {83^ \circ }$
So as we know that the table that is given below:
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So here we know the value of $\tan {60^ \circ }$but we don’t know the value of $\tan {7^ \circ },\tan {23^ \circ },\tan {67^ \circ },\tan {83^ \circ }$
Now we can convert any two of the $\tan {7^ \circ }\,\,$or $\tan {23^ \circ }$ into $\cot \theta $ by using the formula $\tan \theta = \cot \left( {90 - \theta } \right)$
So if $\theta = {7^ \circ }$, then
$
  \tan 7 = \cot \left( {90 - 7} \right) \\
  \tan 7 = \cot \left( {83} \right) \\
 $
So if $\theta = {23^ \circ }$, then
$
  \tan 23 = \cot \left( {90 - 23} \right) \\
  \tan 23 = \cot \left( {67} \right) \\
 $
So we need to find the value of $\tan {7^ \circ }\tan {23^ \circ }\tan {60^ \circ }\tan {67^ \circ }\tan {83^ \circ }$
Now replace $\tan {7^ \circ }\,\,$and $\tan {23^ \circ }$ with $\cot {83^{ \circ \,}}\,\,\,$and $\cot {67^ \circ }$ respectively.
So we will get, $\cot {83^ \circ }\cot {67^ \circ }\tan {60^ \circ }\tan {67^ \circ }\tan {83^ \circ }$
Now after rearranging
\[(\cot {83^ \circ }\tan {83^ \circ })\tan {60^ \circ }(\tan {67^ \circ }\cot {67^ \circ })\]
We know that $\tan \theta \cot \theta = 1$. So using we get
$(\cot {83^ \circ }\tan {83^ \circ }) = 1$
And \[(\tan {67^ \circ }\cot {67^ \circ }) = 1\]
Putting these value we get,
\[(1) \times \tan {60^ \circ } \times (1)\]
And we know that \[\tan {60^ \circ } = \sqrt 3 \]
So we get the product of $\tan {7^ \circ }\tan {23^ \circ }\tan {60^ \circ }\tan {67^ \circ }\tan {83^ \circ }$$ = \sqrt 3 $

So, the correct answer is “Option B”.

Note:We should learn standard trigonometric angles of $\sin \theta ,\cos \theta \,\& \tan \theta $.
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And we should know the relations $\tan \theta . \cot \theta = 1,\cos \theta . \sec \theta = 1, \cos ec \theta . \sin \theta = 1$.Students should also remember trigonometric formulas and identities for solving these types of problems.