Question

The value of $ \cos y\cos \left( {\dfrac{\pi }{2} - x} \right) - \cos \left( {\dfrac{\pi }{2} - y} \right)\cos x + \sin y\cos \left( {\dfrac{\pi }{2} - x} \right) + \cos x\sin \left( {\dfrac{\pi }{2} - y} \right) $ is zero if
(A) $ x = 0 $
(B) $ y = 0 $
(C) $ x = y $
(D) $ x = n\pi - \dfrac{\pi }{4} + y,n \in I $

Answer 1

Hint: In the given question, we are provided with the value of an expression involving trigonometric functions. So, we have to find the value of x and y. The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $ \cos \left( {\dfrac{\pi }{2} - x} \right) = \sin x $ and $ \sin \left( {\dfrac{\pi }{2} - y} \right) = \cos y $ . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step-by-step answer:
In the given problem, we are given that,
 $ \cos y\cos \left( {\dfrac{\pi }{2} - x} \right) - \cos \left( {\dfrac{\pi }{2} - y} \right)\cos x + \sin y\cos \left( {\dfrac{\pi }{2} - x} \right) + \cos x\sin \left( {\dfrac{\pi }{2} - y} \right) = 0 $
Now, we know that sine and cosine trigonometric functions are complementary functions. So, we use the trigonometric formulae $ \cos \left( {\dfrac{\pi }{2} - x} \right) = \sin x $ and $ \sin \left( {\dfrac{\pi }{2} - y} \right) = \cos y $ in the given expression. So, we get,
 $ \Rightarrow \cos y\sin x - \sin y\cos x + \sin y\sin x + \cos x\cos y = 0 $
Now, we group the trigonometric terms in a systematic order. So, we get,
 $ \Rightarrow \left( {\sin x\cos y - \cos x\sin y} \right) + \left( {\cos x\cos y + \sin y\sin x} \right) = 0 $
Now, we know the compound angle formulae for sine and cosine trigonometric functions. So, we use the trigonometric formulae $ \sin x\cos y - \cos x\sin y = \sin \left( {x - y} \right) $ and $ \cos x\cos y + \sin y\sin x = \cos \left( {x - y} \right) $ . So, we get,
 $ \Rightarrow \sin \left( {x - y} \right) + \cos \left( {x - y} \right) = 0 $
Now, we shift the terms in the equation and find the value of the tangent of the angle. So, we get,
 $ \Rightarrow \sin \left( {x - y} \right) = - \cos \left( {x - y} \right) $
 $ \Rightarrow \dfrac{{\sin \left( {x - y} \right)}}{{\cos \left( {x - y} \right)}} = - 1 $
 $ \Rightarrow \tan \left( {x - y} \right) = - 1 $
 $ \Rightarrow \tan \left( {x - y} \right) = \tan \left( { - \dfrac{\pi }{4}} \right) $
So, we get the equation in $ \tan \left( A \right) = \tan \left( B \right) $ form. So, the general solution of this equation is of the form $ A = n\pi + B $ , where n is any integer.
So, we have, $ \tan \left( {x - y} \right) = \tan \left( { - \dfrac{\pi }{4}} \right) $ .
Hence, $ x - y = n\pi + \left( { - \dfrac{\pi }{4}} \right) $ , where n is an integer.
So, we get the value of x as $ x = n\pi + \left( { - \dfrac{\pi }{4}} \right) + y,n \in I $
Hence, option (D) is the correct answer.
So, the correct answer is “Option D”.

Note: Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: $ \sin \left( {\dfrac{\pi }{2} - y} \right) = \cos y $ and $ \tan (x) = \dfrac{{\sin (x)}}{{\cos (x)}} $ . Besides these simple trigonometric formulae, we should remember the formats of general trigonometric solutions such as $ \tan \left( A \right) = \tan \left( B \right) $ . Take care while doing the calculations so as to be sure of the final answer.