What is the value of $ an {{ ext{5}}^0} an 2{{ ext{5}}^0} an {30^0} an 6{{ ext{5}}^0} an 8{{ ext{5}}^0} = $$ \left( a ight){ ext{ 1}} \ \left( b ight){ ext{ }}\dfrac{1}{2} \ \left( c ight){ ext{ }}\sqrt 3 \ \left( d ight){ ext{ }}\dfrac{1}{{\sqrt 3 }} \ $

Question

What is the value of $	an {{	ext{5}}^0}	an 2{{	ext{5}}^0}	an {30^0}	an 6{{	ext{5}}^0}	an 8{{	ext{5}}^0} = $$  \left( a ight){	ext{ 1}}   \  \left( b ight){	ext{ }}\dfrac{1}{2}   \  \left( c ight){	ext{ }}\sqrt 3    \  \left( d ight){	ext{ }}\dfrac{1}{{\sqrt 3 }}   \  $

Vedantu Content Team · Accepted Answer

Hint- Use the basic trigonometric identity of $\tan (90 - \theta ) = \cot \theta $
Now we have to find the value of $\tan {{\text{5}}^0}\tan 2{{\text{5}}^0}\tan {30^0}\tan
6{{\text{5}}^0}\tan 8{{\text{5}}^0}$
Using $\tan {30^0} = \dfrac{1}{{\sqrt 3 }}$above we get
\[{\text{tan}}{{\text{5}}^0}{\text{tan2}}{{\text{5}}^0}\left( {\dfrac{1}{{\sqrt 3 }}}
\right){\text{tan6}}{{\text{5}}^0}{\text{tan8}}{{\text{5}}^0}\]
Now we can write \[{\text{tan6}}{{\text{5}}^0}\]as \[{\text{tan}}\left( {90 - 25} \right)\]and similar concept we will to \[{\text{tan8}}{{\text{5}}^0}\]
Thus we get
\[{\text{tan}}{{\text{5}}^0}{\text{tan2}}{{\text{5}}^0}\left( {\dfrac{1}{{\sqrt 3 }}} \right){\text{tan}}\left(
{90 - 25} \right){\text{tan}}\left( {90 - 5} \right)\]
Using the concept that $\tan (90 - \theta ) = \cot \theta $ we can rewrite the above as
\[{\text{tan}}{{\text{5}}^0}{\text{tan2}}{{\text{5}}^0}\left( {\dfrac{1}{{\sqrt 3 }}}
\right){\text{cot2}}{{\text{5}}^0}\cot {5^0}\]
As $\tan \theta = \dfrac{1}{{\cot \theta }}$
The above equation is simplified to \[\left( {\dfrac{1}{{\sqrt 3 }}} \right)\]
So option (d) is the right answer.

Note- The key concept that we need to recall every time we solve such type of problem is that always try
and convert one angle into other by subtracting or even sometimes adding it with the number that can
help changing the trigonometric term in order to cancel them with other terms to reach to the
simplified answer.

What is the value of $\tan {{\text{5}}^0}\tan 2{{\text{5}}^0}\tan {30^0}\tan 6{{\text{5}}^0}\tan
8{{\text{5}}^0} = $
$
  \left( a \right){\text{ 1}} \\
  \left( b \right){\text{ }}\dfrac{1}{2} \\
  \left( c \right){\text{ }}\sqrt 3 \\
  \left( d \right){\text{ }}\dfrac{1}{{\sqrt 3 }} \\
$

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What is the value of $\tan {{\text{5}}^0}\tan 2{{\text{5}}^0}\tan {30^0}\tan 6{{\text{5}}^0}\tan 8{{\text{5}}^0} = $$ \left( a \right){\text{ 1}} \\ \left( b \right){\text{ }}\dfrac{1}{2} \\ \left( c \right){\text{ }}\sqrt 3 \\ \left( d \right){\text{ }}\dfrac{1}{{\sqrt 3 }} \\ $

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NCERT Book Solutions

Reference Book Solutions

Question Papers

Exams

Quick Links

What is the value of $\tan {{\text{5}}^0}\tan 2{{\text{5}}^0}\tan {30^0}\tan 6{{\text{5}}^0}\tan
8{{\text{5}}^0} = $
$
\left( a \right){\text{ 1}} \\
\left( b \right){\text{ }}\dfrac{1}{2} \\
\left( c \right){\text{ }}\sqrt 3 \\
\left( d \right){\text{ }}\dfrac{1}{{\sqrt 3 }} \\
$