Question

The value of $ \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) $ is ?

Answer 1

Hint: The given problem requires us to simplify the given trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities while solving such questions.

Complete step-by-step answer:
In the given question, we are required to evaluate the value of $ \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) $ using the basic concepts of trigonometry and identities.
We can simplify the given trigonometric expression using the trigonometric identity of complementary angles and trigonometric ratios.
So, we have, $ \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) $
Now, we can see that the angles give to us in the question are $ \left( {\dfrac{\pi }{8}} \right) $ and $ \left( {\dfrac{{3\pi }}{8}} \right) $ . So, we can observe that the sum of both the angles is $ \left( {\dfrac{\pi }{8}} \right) + \left( {\dfrac{{3\pi }}{8}} \right) = \left( {\dfrac{{4\pi }}{8}} \right) = \left( {\dfrac{\pi }{2}} \right) $ .
Hence, both the angles are complementary angles as they sum up to $ \left( {\dfrac{\pi }{2}} \right) $ .
So, we have, $ \left( {\dfrac{{3\pi }}{8}} \right) = \left( {\dfrac{\pi }{2}} \right) - \left( {\dfrac{\pi }{8}} \right) $ .
Hence, we get, $ \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) = \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{\pi }{2} - \dfrac{\pi }{8}} \right) $
Now, we know that tangent and cotangent are complementary ratios of each other. This means that $ \cot \left( x \right) = \tan \left( {\dfrac{\pi }{2} - x} \right) $ .
 $ \Rightarrow \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) = \tan \left( {\dfrac{\pi }{8}} \right) \times \cot \left( {\dfrac{\pi }{8}} \right) $
Now, we also know that tangent and cotangent are reciprocal ratios of each other. So, we get the product of tangent and cotangent of the same angle as one.
Hence, we have,
 $ \Rightarrow \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) = 1 $
So, we have simplified the trigonometric expression $ \tan \left( {\dfrac{\pi }{8}} \right) \times \tan \left( {\dfrac{{3\pi }}{8}} \right) $ by using the complementary angle formula and the fact that the trigonometric ratios tangent and cotangent are reciprocals of each other as $ 1 $ .
So, the correct answer is “1”.

Note: There are six trigonometric ratios: $ \sin \theta $ , $ \cos \theta $ , $ \tan \theta $ , $ \cos ec\theta $ , $ \sec \theta $ and $ \cot \theta $ . All the trigonometric ratios can be converted into each other using the simple trigonometric identities. The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.