Question

Evaluate the given expression $\tan {43^0}\tan {45^0}\tan {47^0}$.
$
  (a){\text{ }}\sqrt 3 \\
  (b){\text{ }}\dfrac{1}{{\sqrt 3 }} \\
  (c){\text{ 1}} \\
  (d){\text{ 2}} \\
 $

Answer 1

Hint – Use the concept of trigonometric conversion that $\tan \left( {{{90}^0} - \theta } \right) = \cot \theta $, in order to neutralize the terms of the given expression as we know that $\cot \theta = \dfrac{1}{{\tan \theta }}$. Trigonometric term remaining will be standard value based so directly use its value.

Complete step-by-step answer:

Given trigonometric equation is
$\tan {43^0}\tan {45^0}\tan {47^0}$

The above equation is also written as
$ \Rightarrow \tan {43^0}\tan {45^0}\tan \left( {{{90}^0} - {{43}^0}} \right)$

Now as we know the basic trigonometric property that $\tan \left( {{{90}^0} - \theta } \right) = \cot \theta $, so use this property in above equation we have,
$ \Rightarrow \tan {43^0}\tan {45^0}\cot {43^0}$

Now again use the basic trigonometric identity that $\cot \theta = \dfrac{1}{{\tan \theta }}$ , so use this property in above equation we have,
$ \Rightarrow \tan {43^0}\tan {45^0}\dfrac{1}{{\tan {{43}^0}}}$

Now as we see $\tan {43^0}$ is cancel out so the remaining term is
$ \Rightarrow \tan {43^0}\tan {45^0}\dfrac{1}{{\tan {{43}^0}}} = \tan {45^0}$

Now as we know that the value of $\tan {45^0}$ is 1.
$ \Rightarrow \tan {43^0}\tan {45^0}\tan {47^0} = 1$
So this is the required answer & option (C) is correct.

Note – The basic understanding of trigonometric identities helps in solving problems of this kind. It is advised to remember these identities as it helps save a lot of time, with practice the identities and its application becomes more transparent in mind, so keep practicing.