Question

Find the value of $ \sqrt {4{{\cos }^4}\theta + {{\sin }^2}2\theta } + 4\cot \theta {\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$, if \[\dfrac{\pi }{2} < \theta < \dfrac{{3\pi }}{2}\].
A) $2\sin \theta $
B) $ - 2\sin \theta $
C) $2\cot \theta $
D) $ - 2\cot \theta $

Answer 1

Hint: we can solve this problem by the simplifying given term.
Here we can use some trigonometric formulas.
1) $\sin 2\theta = 2\sin \theta \cos \theta $
2) $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$
3) $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$
4) ${\sin ^2}\theta + {\cos ^2}\theta = 1$

Complete step by step answer:
1) Let us simplify the given term.
We have,
 $ = \sqrt {4{{\cos }^4}\theta + {{\sin }^2}2\theta } + 4\cot \theta {\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
Let us simplify first the terms in the square root.
We can use 1) formula.
We get,
$ \Rightarrow \sqrt {4{{\cos }^4}\theta + {{\left( {2\sin \theta \cos \theta } \right)}^2}} + 4\cot \theta {\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
Simplify the remaining term.
Let us use 2) formula.
Now we get,
$ \Rightarrow \sqrt {4{{\cos }^4}\theta + {{\left( {2\sin \theta \cos \theta } \right)}^2}} + 4\cot \theta {\left[ {\cos \dfrac{\pi }{4}\cos \dfrac{\theta }{2} + \sin \dfrac{\pi }{4}\sin \dfrac{\theta }{2}} \right]^2}$
Now, we can put the values of cos and sin.
As we know the values of, $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and$\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$.
Substitute these values.
$ \Rightarrow \sqrt {4{{\cos }^4}\theta + {{\left( {2\sin \theta \cos \theta } \right)}^2}} + 4\cot \theta {\left[ {\dfrac{1}{{\sqrt 2 }}\cos \dfrac{\theta }{2} + \dfrac{1}{{\sqrt 2 }}\sin \dfrac{\theta }{2}} \right]^2}$
Consider the term which is under the square root and is in the bracket.
Take the square of that term.
See, $\dfrac{1}{{\sqrt 2 }}$ is in both the terms.
So, let us write it outside the bracket as a common term.
So we get,
$ \Rightarrow \sqrt {4{{\cos }^4}\theta + \left( {4{{\sin }^2}\theta {{\cos }^2}\theta } \right)} + \dfrac{4}{{{{\left( {\sqrt 2 } \right)}^2}}}\cot \theta {\left[ {\cos \left( {\dfrac{\theta }{2}} \right) + \sin \left( {\dfrac{\theta }{2}} \right)} \right]^2}$
Consider the term which is under the square root.
Let us take common terms outside the bracket.
So we get,
$ \Rightarrow \sqrt {4{{\cos }^2}\theta \left( {{{\sin }^2}\theta + {{\cos }^2}\theta } \right)} + \dfrac{4}{{{{\left( {\sqrt 2 } \right)}^2}}}\cot \theta {\left[ {\cos \left( {\dfrac{\theta }{2}} \right) + \sin \left( {\dfrac{\theta }{2}} \right)} \right]^2}$
While removing the square root so, the value which is in the square root will be in the mod.
After that, we will use the formula to simplify the bracket:${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
$ \Rightarrow \left| {2\cos \theta \left( 1 \right)} \right| + \dfrac{4}{2}\cot \theta \left[ {{{\cos }^2}\left( {\dfrac{\theta }{2}} \right) + {{\sin }^2}\left( {\dfrac{\theta }{2}} \right) + 2\sin \left( {\dfrac{\theta }{2}} \right)\cos \left( {\dfrac{\theta }{2}} \right)} \right]$
Use the fourth formula which is in the hint.
$ \Rightarrow \left| {2\cos \theta } \right| + 2\cot \theta \left[ {1 + \sin \theta } \right]$
Simplify the bracket.
$ \Rightarrow \left| {2\cos \theta } \right| + 2\cot \theta + 2\cot \theta \sin \theta $
As we know, $\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$.
$ \Rightarrow \left| {2\cos \theta } \right| + 2\cot \theta + 2\dfrac{{\cos \theta }}{{\sin \theta }}\sin \theta $
Here, $\sin \theta $ get cancelled.
$ \Rightarrow 2\left[ {\left| {\cos \theta } \right| + \cot \theta + \cos \theta } \right]$
In both the 2 & 3 quadrants we have given the value of cos is negative.
As the first cos is in the mod. So, there is no need to change the sign of that term.
$ \Rightarrow 2\left[ {\left| {\cos \theta } \right| + \cot \theta - \cos \theta } \right]$
$ \Rightarrow 2\left[ {\cot \theta } \right]$
$ \Rightarrow 2\cot \theta $

Therefore, option (C) is the correct answer.
Note:
 - To solve this kind of problem, students should be familiar with trigonometric basic formulas, identities, reciprocal identities, angle sum & difference identities.
- Students should know about trigonometric functions and the signs of main trigonometric functions sin, cos, tan in different quadrants.
- Important thing is that students should know the values of all trigonometric functions at various degrees.