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.