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Find the value of $\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ$.(a) $\dfrac{1}{16}$(b) $0$ (c) $1$(d) None of these

Last updated date: 20th Mar 2023
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Hint: First use the formula of $\tan \theta$ to $\cot \theta$ conversion which is: $\tan \left( 90{}^\circ -\theta \right)=\cot \theta$.
Then use the formula $tan\theta \cot \theta =1$

We need to find the value of $\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ$
We will first use the formula of $\tan \theta$ to $\cot \theta$ conversion: on either the first two terms or the last two terms.

Here, let us use it on $\tan 10{}^\circ$and $\tan 15{}^\circ$.
So $\tan 10{}^\circ$can be written as $\tan \left( 90{}^\circ -80{}^\circ \right)$

Now use the formula $\tan \left( 90{}^\circ -\theta \right)=\cot \theta$ where $\theta =80{}^\circ$

We will get the following:
$\tan \left( 90{}^\circ -80{}^\circ \right)=\cot 80{}^\circ$
So,$\tan 10{}^\circ =\cot 80{}^\circ$ …(1)

Similarly, $\tan 15{}^\circ$ can be written as $\tan \left( 90{}^\circ -75{}^\circ \right)$
Now use the formula $\tan \left( 90{}^\circ -\theta \right)=\cot \theta$ where $\theta =75{}^\circ$

We will get the following:
$\tan \left( 90{}^\circ -75{}^\circ \right)=\cot 75{}^\circ$
So, $\tan 15{}^\circ =\cot 75{}^\circ$ …(2)

Now we will substitute equations (1) and (2) in the given expression:
$\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ =\cot 80{}^\circ \cot 75{}^\circ \tan 75{}^\circ \tan 80{}^\circ$

Rearranging the terms, we get the following:
$\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ =\left( \tan 80{}^\circ \cot 80{}^\circ \right)\left( \tan 75{}^\circ \cot 75{}^\circ \right)$ …(3)

Now we will use the formula $tan\theta \cot \theta =1$ which is derived by using $\left( \cot \theta =\dfrac{1}{\tan \theta } \right)$
So, $\tan 80{}^\circ \cot 80{}^\circ =1$
And $\tan 75{}^\circ \cot 75{}^\circ =1$ …(4)

We will now substitute the equations in (4) to (3), we will get the following:
$\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ =\left( \tan 80{}^\circ \cot 80{}^\circ \right)\left( \tan 75{}^\circ \cot 75{}^\circ \right)$
$\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ =\left( 1 \right)\left( 1 \right)$
So, $\tan 10{}^\circ \tan 15{}^\circ \tan 75{}^\circ \tan 80{}^\circ =1$
which is option (c)

So the final answer is (c) $1$

Note: Instead of converting the first two terms from $\tan 10{}^\circ$and $\tan 15{}^\circ$to $\cot 80{}^\circ$and $\cot 75{}^\circ$respectively, one can change the last two terms from $\tan 75{}^\circ$and $\tan 80{}^\circ$to $\cot 15{}^\circ$and $\cot 10{}^\circ$respectively too and then proceed. The final answer will be the same in both the cases.