Answer
Verified
411.3k+ views
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.
Complete step-by-step answer:
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.
Complete step-by-step answer:
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.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE