Question

If \[\sin \left( {\dfrac{\pi }{4}\cot \theta } \right) = \cos \left( {\dfrac{\pi }{4}\tan \theta } \right)\] then \[\theta = n\pi + \dfrac{\pi }{4}\] , \[n \in Z\]
II. \[\tan \left( {\dfrac{\pi }{2}\sin \theta } \right) = \cot \left( {\dfrac{\pi }{2}\cos \theta } \right)\] then \[\sin \left( {\theta + \dfrac{\pi }{4}} \right) = \pm \dfrac{1}{{\sqrt 2 }}\]
A.Only I is true
B.Only II is true
C.Both I and II are true
D.Neither I or II are true

Answer 1

Hint: We are asked to find out which of the following statements are true. For this try to find out the value of \[\theta \] for each statement. Equate L.H.S to R.H.S such that you can simply find the value of \[\theta \] . Then compare with the options given and select the appropriate answer.

Complete step-by-step answer:
Given, two expressions
 \[\sin \left( {\dfrac{\pi }{4}\cot \theta } \right) = \cos \left( {\dfrac{\pi }{4}\tan \theta } \right)\] and \[\tan \left( {\dfrac{\pi }{2}\sin \theta } \right) = \cot \left( {\dfrac{\pi }{2}\cos \theta } \right)\]

Let us check each expression one by one.
The first expression is \[\sin \left( {\dfrac{\pi }{4}\cot \theta } \right) = \cos \left( {\dfrac{\pi }{4}\tan \theta } \right)\]
 \[L.H.S = \sin \left( {\dfrac{\pi }{4}\cot \theta } \right)\] (i)
We know, \[\sin \theta = \cos \left( {\dfrac{\pi }{2} - \theta } \right)\]
Using this in equation (i), we get
 \[L.H.S = \cos \left( {\dfrac{\pi }{2} - \dfrac{\pi }{4}\cot \theta } \right)\]
Equating with the R.H.S we get
 \[\cos \left( {\dfrac{\pi }{2} - \dfrac{\pi }{4}\cot \theta } \right) = \cos \left( {\dfrac{\pi }{4}\tan \theta } \right)\]
Equating the angles of cosine, we get
 \[\Rightarrow \dfrac{\pi }{2} = \dfrac{\pi }{4}\tan \theta + \dfrac{\pi }{4}\cot \theta \\
   \Rightarrow \dfrac{\pi }{2} = \dfrac{\pi }{4}(\tan \theta + \cot \theta ) \\
   \Rightarrow \dfrac{\pi }{2} = \dfrac{\pi }{4}\left( {\tan \theta + \dfrac{1}{{\tan \theta }}} \right) \\
   \Rightarrow \dfrac{\pi }{2} = \dfrac{\pi }{4}\left( {\dfrac{{1 + {{\tan }^2}\theta }}{{\tan \theta }}} \right) \]
We know, \[1 + {\tan ^2}\theta = {\sec ^2}\theta \] using this we get
 \[\dfrac{\pi }{2} = \dfrac{\pi }{4}\left( {\dfrac{{{{\sec }^2}\theta }}{{\tan \theta }}} \right) \\
   \Rightarrow 2\tan \theta = {\sec ^2}\theta \\
   \Rightarrow 2\left( {\dfrac{{\sin \theta }}{{\cos \theta }}} \right) = \left( {\dfrac{1}{{{{\cos }^2}\theta }}} \right) \]
 \[\Rightarrow 2\sin \theta \cos \theta = 1 \\
   \Rightarrow \sin 2\theta = 1 \\
   \Rightarrow 2\theta = \dfrac{{n\pi }}{2} \\
   \Rightarrow \theta = \dfrac{{n\pi }}{4} \]
Where n is any integer
Therefore statement I is incorrect as it says \[\theta = n\pi + \dfrac{\pi }{4}\] but \[\theta = \dfrac{{n\pi }}{4}\] .
Second expression: \[\tan \left( {\dfrac{\pi }{2}\sin \theta } \right) = \cot \left( {\dfrac{\pi }{2}\cos \theta } \right)\]
We can write \[\cot \left( {\dfrac{\pi }{2} - \theta } \right) = \tan \theta \] .
Using this we get
 \[\cot \left( {\dfrac{\pi }{2} - \dfrac{\pi }{2}\sin \theta } \right) = \cot \left( {\dfrac{\pi }{2}\cos \theta } \right)\]
Equating the angles we get,
 \[\left( {\dfrac{\pi }{2} - \dfrac{\pi }{2}\sin \theta } \right) = \left( {\dfrac{\pi }{2}\cos \theta } \right)\]
 \[\Rightarrow \dfrac{\pi }{2} = \dfrac{\pi }{2}\sin \theta + \dfrac{\pi }{2}\cos \theta \\
   \Rightarrow \sin \theta + \cos \theta = 1 \]
Multiplying both sides by \[\dfrac{1}{{\sqrt 2 }}\] , we get
 \[\dfrac{1}{{\sqrt 2 }}\sin \theta + \dfrac{1}{{\sqrt 2 }}\cos \theta = \dfrac{1}{{\sqrt 2 }}\]
We know, \[\cos \dfrac{\pi }{4} = \sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\] using this we get
 \[\sin \theta \cos \dfrac{\pi }{4} + \cos \theta \sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
We have \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] , using this we get
 \[\sin \left( {\theta + \dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}\]
Therefore, statement II is true.
Hence, the correct answer is option (B) Only II is true
So, the correct answer is “Option B”.

Note: For such questions, where the value of \[\theta \] is given and you need to check whether it is true or not, try to simplify the expression such that you can find the value of \[\theta \] . Most importantly, always remember the trigonometric identities as these help us to get a simplified answer.
I. If \[\sin \left( {\dfrac{\pi }{4}\cot \theta } \right) = \cos \left( {\dfrac{\pi }{4}\tan \theta } \right)\] then \[\theta = n\pi + \dfrac{\pi }{4}\] , \[n \in Z\]
II. \[\tan \left( {\dfrac{\pi }{2}\sin \theta } \right) = \cot \left( {\dfrac{\pi }{2}\cos \theta } \right)\] then \[\sin \left( {\theta + \dfrac{\pi }{4}} \right) = \pm \dfrac{1}{{\sqrt 2 }}\]