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If trigonometric equation is given as $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$ then $\cos (\theta - \dfrac{\pi }{4})$ is equal to$A.{\text{ }} \pm \dfrac{1}{{2\sqrt 2 }} \\ B.{\text{ }} \pm \dfrac{1}{{\sqrt 2 }} \\ C.{\text{ }} \pm \sqrt 2 \\ D.{\text{ }} \pm 2\sqrt 2 \\$

Last updated date: 24th Mar 2023
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Hint- In order to solve this question we will use the simple trigonometric identities such as $\tan ({90^0} - \theta ) = \cot \theta$ and $\cos (A - B) = \cos A\cos B + \sin A\sin B.$ So we will try to make the given term in this form to proceed further.

Complete step-by-step solution -
Given that $\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$
Now, proceeding further with the given equation
$\Rightarrow \tan (\pi \cos \theta ) = \cot (\pi \sin \theta )$
As we know that $\left[ {\cot A = \tan (\dfrac{\pi }{2} - A)} \right]$
Using the above formula in the given equation, we get
$\Rightarrow \tan (\pi \cos \theta ) = \tan ( \pm \dfrac{\pi }{2} - \pi \sin \theta ) \\ \Rightarrow \pi \cos \theta = \pm \dfrac{\pi }{2} - \pi \sin \theta \\ \Rightarrow \pi (\cos \theta + \sin \theta ) = \pm \dfrac{\pi }{2} \\ \Rightarrow (\cos \theta + \sin \theta ) = \pm \dfrac{1}{2} \\$
Now, we will multiply LHS by $\dfrac{{\sqrt 2 }}{{\sqrt 2 }}$ to from a cosine formula
$\Rightarrow \dfrac{{\sqrt 2 }}{{\sqrt 2 }}(\cos \theta + \sin \theta ) = \pm \dfrac{1}{2} \\ \Rightarrow \sqrt 2 (\dfrac{{\cos \theta }}{{\sqrt 2 }} + \dfrac{{\sin \theta }}{{\sqrt 2 }}) = \pm \dfrac{1}{2} \\$
Since, we know that$\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }} = \sin \dfrac{\pi }{4}$ , substituting this in the above formula
$\Rightarrow \sqrt 2 (\cos \theta \cos \dfrac{\pi }{4} + \sin \theta \sin \dfrac{\pi }{4}) = \pm \dfrac{1}{2}$
As we know that$\left[ {\cos (A - B) = \cos A\cos B + \sin A\sin B} \right]$
$\Rightarrow \cos (\theta - \dfrac{\pi }{4}) = \pm \dfrac{1}{{2\sqrt 2 }}$
Hence, the correct option is A

Note- In such type of questions starts solving from the complex side of the questions and tries to express every term in terms of sin and cosine or a variable which is easy to solve. To simplify these questions try to combine two terms to a single term using trigonometric formulas.