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 + \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 }}$$