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Question: If \(\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )\) then a value of \(\cos (\theta - \dfrac{\pi...

If tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ)\tan (\pi \cos \theta ) = \cot (\pi \sin \theta ) then a value of cos⁡(θ−π4)\cos (\theta - \dfrac{\pi }{4}) among the following is
A. 122\dfrac{1}{{2\sqrt 2 }}
B. 12\dfrac{1}{{\sqrt 2 }}
C. 12\dfrac{1}{2}
D. 14\dfrac{1}{4}

Explanation

Solution

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⁡(θ−π4)\cos (\theta - \dfrac{\pi }{4}) by using standard formula and then the value is computed.

Complete step-by-step answer:
Given equation is tan⁡(πcos⁡θ)=cot⁡(πsin⁡θ)\tan (\pi \cos \theta ) = \cot (\pi \sin \theta ).
We know that cot⁡(θ)=tan⁡(π2−θ)\cot (\theta ) = \tan \left( {\dfrac{\pi }{2} - \theta } \right).
Hence, cot⁡(πsin⁡θ)=tan⁡(π2−πsin⁡θ)\cot (\pi \sin \theta ) = \tan \left( {\dfrac{\pi }{2} - \pi \sin \theta } \right).
Substituting above value in given equation,we get
tan⁡(πcos⁡θ)=tan⁡(π2−πsin⁡θ)\tan (\pi \cos \theta ) = \tan \left( {\dfrac{\pi }{2} - \pi \sin \theta } \right)
Now we can cancel out tan from both sides
πcos⁡θ=π2−πsin⁡θ\pi \cos \theta = \dfrac{\pi }{2} - \pi \sin \theta .
On simplifying, we get
cos⁡θ+sin⁡θ=12\cos \theta + \sin \theta = \dfrac{1}{2}.
Multiplying 12\dfrac{1}{{\sqrt 2 }}with above equation we get,
12cos⁡θ+12sin⁡θ=12×12\dfrac{1}{{\sqrt 2 }}\cos \theta + \dfrac{1}{{\sqrt 2 }}\sin \theta = \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }}
As we know cos⁡π4=12\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }} and sin⁡π4=12\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}.
On replacing the equation with above value we get
cos⁡π4.cos⁡θ+sin⁡π4.sin⁡θ=122\cos \dfrac{\pi }{4}.\cos \theta + \sin \dfrac{\pi }{4}.\sin \theta = \dfrac{1}{{2\sqrt 2 }}
We know that cos⁡(θ−π4)=cos⁡θ.cos⁡π4+sin⁡θ.sin⁡π4.→(1)\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⁡(θ−π4)=122\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.