Question

If \[\sin \left( {\pi \cos \theta } \right) = \cos \left( {\pi \sin \theta } \right)\] , then which of the following is correct.
$
  \left( a \right)\cos \theta = \dfrac{3}{{2\sqrt 2 }} \\
  \left( b \right)\cos \left( {\theta - \dfrac{\pi }{2}} \right) = \dfrac{1}{{2\sqrt 2 }} \\
  \left( c \right)\cos \left( {\theta - \dfrac{\pi }{4}} \right) = \dfrac{1}{{2\sqrt 2 }} \\
  \left( d \right)\cos \left( {\theta + \dfrac{\pi }{4}} \right) = - \dfrac{1}{{2\sqrt 2 }} \\
$

Answer 1

Hint-In this question, we use the concept of trigonometric equations. We use the general solution, \[\cos x = \cos y \Rightarrow x = 2n\pi \pm y,n \in I\] . First we try to write a question in terms of the general equation then apply the formula of the trigonometric equation.

Complete step-by-step answer:
We have an equation \[\sin \left( {\pi \cos \theta } \right) = \cos \left( {\pi \sin \theta } \right)\] and we have to write equation in term of general trigonometric equation.
\[\sin \left( {\pi \cos \theta } \right) = \cos \left( {\pi \sin \theta } \right)\]
As we know, \[\sin \theta = \cos \left( {\dfrac{\pi }{2} - \theta } \right)\]
\[ \Rightarrow \cos \left( {\dfrac{\pi }{2} - \pi \cos \theta } \right) = \cos \left( {\pi \sin \theta } \right)\]
Now, we apply \[\cos x = \cos y \Rightarrow x = 2n\pi \pm y,n \in I\]
$ \Rightarrow \dfrac{\pi }{2} - \pi \cos \theta = 2n\pi \pm \pi \sin \theta ,n \in I$
This relation holds for any integer but now we use n=0 for easy calculation.
$
   \Rightarrow \dfrac{\pi }{2} - \pi \cos \theta = \pm \pi \sin \theta \\
   \Rightarrow \dfrac{1}{2} - \cos \theta = \pm \sin \theta \\
   \Rightarrow \dfrac{1}{2} = \cos \theta \pm \sin \theta \\
$
Now, we multiply by $\dfrac{1}{{\sqrt 2 }}$ on both sides of the equation.
\[ \Rightarrow \dfrac{1}{{\sqrt 2 }} \times \cos \theta \pm \dfrac{1}{{\sqrt 2 }} \times \sin \theta = \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2}\]
As we know, \[\cos \dfrac{\pi }{4} = \sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
\[ \Rightarrow \cos \dfrac{\pi }{4}\cos \theta \pm \sin \dfrac{\pi }{4}\sin \theta = \dfrac{1}{{2\sqrt 2 }}\]
If we take negative value, \[\cos \dfrac{\pi }{4}\cos \theta - \sin \dfrac{\pi }{4}\sin \theta = \dfrac{1}{{2\sqrt 2 }}\]
Now we use trigonometric identity, $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$
\[ \Rightarrow \cos \left( {\theta + \dfrac{\pi }{4}} \right) = \dfrac{1}{{2\sqrt 2 }}\]
If we take positive value, \[\cos \dfrac{\pi }{4}\cos \theta + \sin \dfrac{\pi }{4}\sin \theta = \dfrac{1}{{2\sqrt 2 }}\]
 Now we use trigonometric identity, $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$
\[ \Rightarrow \cos \left( {\theta - \dfrac{\pi }{4}} \right) = \dfrac{1}{{2\sqrt 2 }}\]
So, the correct option is (c).
Note-In such types of problems we use some important points to solve questions in an easy way. First we use trigonometric general equations and equate angles for integer n=0 for easy calculation. Then we use trigonometric identities to solve the equation. So, we will get the required answer.