Answer

Verified

432k+ views

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.

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.

Recently Updated Pages

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

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE

What organs are located on the left side of your body class 11 biology CBSE