Question

Solve $ \cos \left( {{{\cos }^{ - 1}}\left( { - \dfrac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \dfrac{1}{7}} \right)} \right) $ equals:
(A) $ \left( { - \dfrac{1}{3}} \right) $
(B) $ 0 $
(C) $ \dfrac{1}{3} $
(D) $ \dfrac{4}{9} $

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. We should know that cosine inverse and sine inverse functions are complementary functions.

Complete step-by-step answer:
In the given question, we are required to evaluate the value of $ \cos \left( {{{\cos }^{ - 1}}\left( { - \dfrac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \dfrac{1}{7}} \right)} \right) $ using the basic concepts of trigonometry and identities.
So, we have, $ \cos \left( {{{\cos }^{ - 1}}\left( { - \dfrac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \dfrac{1}{7}} \right)} \right) $
We know that sine inverse and cosine inverse functions are complementary functions of each other. So, we can say that $ {\cos ^{ - 1}}x + {\sin ^{ - 1}}x = \left( {\dfrac{\pi }{2}} \right) $ . So, we get,
 $ \Rightarrow \cos \left( {\dfrac{\pi }{2}} \right) $
So, the trigonometric expression $ \cos \left( {{{\cos }^{ - 1}}\left( { - \dfrac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \dfrac{1}{7}} \right)} \right) $ is simplified to the expression $ \cos \left( {\dfrac{\pi }{2}} \right) $ . Now, we know that the value of cosine for angle $ \left( {\dfrac{\pi }{2}} \right) $ is zero. So, we get,
 $ \Rightarrow 0 $
Hence, we get the value of trigonometric expression $ \cos \left( {{{\cos }^{ - 1}}\left( { - \dfrac{1}{7}} \right) + {{\sin }^{ - 1}}\left( { - \dfrac{1}{7}} \right)} \right) $ as $ 0 $ .
So, the correct answer is “Option B”.

Note: We must keep in mind the trigonometric formulae and identities in order to solve the given trigonometric equation. We should know the values of trigonometric functions: $ \sin \theta $ , $ \cos \theta $ , $ \tan \theta $ , $ \cos ec\theta $ , $ \sec \theta $ and $ \cot \theta $ for some basic angles such as $ 0 $ , $ \dfrac{\pi }{4} $ , $ \dfrac{\pi }{3} $ , and $ \dfrac{\pi }{2} $ . Basic trigonometric identities include $ {\sin ^2}\theta + {\cos ^2}\theta = 1 $ , $ {\sec ^2}\theta = {\tan ^2}\theta + 1 $ and $ \cos e{c^2}\theta = {\cot ^2}\theta + 1 $ . These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above. 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 rules come into significant use while solving such problems.