QUESTION

If $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$ then a value of $\cos (\theta - \dfrac{\pi }{4})$ among the following isA. $\dfrac{1}{{2\sqrt 2 }}$B. $\dfrac{1}{{\sqrt 2 }}$C. $\dfrac{1}{2}$D. $\dfrac{1}{4}$

Hint: Here we will simplify the given equation by converting any one of the expression into same trigonometric function i.e converting cot to tan trigonometric function in L.H.S by using the formulae of trigonometry.Simplify the equation further and converting it into $\cos (\theta - \dfrac{\pi }{4})$ by using standard formula and then the value is computed.

Given equation is $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$.
We know that $\cot (\theta ) = \tan \left( {\dfrac{\pi }{2} - \theta } \right)$.
Hence, $\cot (\pi \sin \theta ) = \tan \left( {\dfrac{\pi }{2} - \pi \sin \theta } \right)$.
Substituting above value in given equation,we get
$\tan (\pi \cos \theta ) = \tan \left( {\dfrac{\pi }{2} - \pi \sin \theta } \right)$
Now we can cancel out tan from both sides
$\pi \cos \theta = \dfrac{\pi }{2} - \pi \sin \theta$.
On simplifying, we get
$\cos \theta + \sin \theta = \dfrac{1}{2}$.
Multiplying $\dfrac{1}{{\sqrt 2 }}$with above equation we get,
$\dfrac{1}{{\sqrt 2 }}\cos \theta + \dfrac{1}{{\sqrt 2 }}\sin \theta = \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }}$
As we know $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and $\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$.
On replacing the equation with above value we get
$\cos \dfrac{\pi }{4}.\cos \theta + \sin \dfrac{\pi }{4}.\sin \theta = \dfrac{1}{{2\sqrt 2 }}$
We know that $\cos (\theta - \dfrac{\pi }{4}) = \cos \theta .\cos \dfrac{\pi }{4} + \sin \theta .\sin \dfrac{\pi }{4}. \to (1)$
Therefore, using equation (1) we have
$\cos (\theta - \dfrac{\pi }{4}) = \dfrac{1}{{2\sqrt 2 }}$.
Hence the correct option is A.

Note: In these type of questions we have to know the general formula of trigonometry.Students should remember trigonometric identities and important formulas for solving these type of problems.Try to convert the equations or simplify to standard formula to get the desired answer.We can also convert tan to cot trigonometric function in L.H.S and further simplifying it,we will get same answer.