Question

FInd the value of $\tan {5^ \circ } \times \tan {30^ \circ } \times 4\tan {85^ \circ }$
(A) $\dfrac{4}{{\sqrt 3 }}$
(B) $4\sqrt 3 $
(C) $1$
(D) $4$

Answer 1

Hint: The given question deals with finding the value of trigonometric expression doing basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\cot x = \dfrac{1}{{\tan x}}$ and value of tangent for some standard angles. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem. We must know the simplification rules to solve the problem with ease.

Complete step-by-step solution:
In the given problem, we have to find the value of $\tan {5^ \circ } \times \tan {30^ \circ } \times 4\tan {85^ \circ }$.
So, we know that cotangent and tangent are complementary functions of each other. So, we have, $\tan \left( {{{90}^ \circ } - x} \right) = \cot x$.
Hence, we get,
$= \tan {5^ \circ } \times \tan {30^ \circ } \times 4\cot \left( {{{90}^ \circ } - {{85}^ \circ }} \right)$
$= \tan {5^ \circ } \times \tan {30^ \circ } \times 4\cot {5^ \circ }$
Now, we know that cotangent and tangent trigonometric functions are reciprocal functions of each other.
$= \tan {5^ \circ } \times \tan {30^ \circ } \times \dfrac{4}{{\tan {5^ \circ }}}$
Cancelling the common terms in numerator and denominator, we get,
$= \tan {30^ \circ } \times 4$
Now, we know the value of $\tan {30^ \circ }$ as $\dfrac{1}{{\sqrt 3 }}$. So, we get,
$= \dfrac{4}{{\sqrt 3 }}$
Therefore, the value of $\tan {5^ \circ } \times \tan {30^ \circ } \times 4\tan {85^ \circ }$ is equal to $\dfrac{4}{{\sqrt 3 }}$.
Hence, Option (A) is the correct answer.

Note: We must have a strong grip over the concepts of trigonometry, related formulae and rules to ace these types of questions. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers but we have to mark the most appropriate option among the given choices.