Question

If $\pi < \theta < \dfrac{{3\pi }}{2}$ , the expression $\sqrt {(4{{\sin }^4}\theta + 2{{\sin }^2}2\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$ equal to
A. $2$
B. $2 + 4\sin \theta$
C. $2 - 4\sin \theta$
D. None of these

Answer 1

Here we are asked to find the value of the given expression. But in the given expression there are trigonometric functions with higher powers. So, we need to use the existing trigonometric formula and simplify it. By doing this we can find the value of the given expression.
Formula: Some of the formulas that we need to know to solve this problem are:
$\cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta$
$\sin (2x) = 2\sin x\cos x$
${\sin ^2}x + {\cos ^2}x = 1$
$\cos (2x) = 2{\cos ^2}x - 1$
$\sqrt {{a^2}} = a$

Complete step by step answer:
It is given that the angle lies between $\pi \& \dfrac{{3\pi }}{2}$ . The sine function is negative in this range. We aim to find the value of the expression $\sqrt {(4{{\sin }^4}\theta + 2{{\sin }^2}2\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$ .
Now let us simplify the part inside the square root using the formula $\sin (2x) = 2\sin x\cos x$ .
$\Rightarrow$ $\sqrt {(4{{\sin }^4}\theta + 2[2{{\sin }^2}\theta {{\cos }^2}\theta ])} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
On further simplification we get
$\Rightarrow$ $\sqrt {(4{{\sin }^4}\theta + 4{{\sin }^2}\theta {{\cos }^2}\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
Now let us take $4{\sin ^2}\theta$ commonly out.
$\Rightarrow$ $\sqrt {{2^2}{{\sin }^2}\theta ({{\sin }^2}\theta + {{\cos }^2}\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
Let’s modify the above expression.
$\Rightarrow$ $\sqrt {{2^2}{{\sin }^2}\theta ({{\sin }^2}\theta + {{\cos }^2}\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$
Now using the formula $\sqrt {{a^2}} = a$ we get
$\Rightarrow$ $2\sqrt {{{\sin }^2}\theta ({{\sin }^2}\theta + {{\cos }^2}\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$ ………. $(1)$
Considering the formula $\cos (2x) = 2{\cos ^2}x - 1$ we get
$\cos (2\theta ) = 2{\cos ^2}\theta - 1 \Rightarrow 2{\cos ^2}\theta = 1 + \cos (2\theta )$
Therefore, $2{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right) = 1 + \cos \left( {\dfrac{{2\pi }}{4} - \dfrac{{2\theta }}{2}} \right) = 1 + \cos \left( {\dfrac{\pi }{2} - \theta } \right)$
Now let’s substitute it in the equation $(1)$ .
$\Rightarrow$ $2\sqrt {{{\sin }^2}\theta ({{\sin }^2}\theta + {{\cos }^2}\theta )} + 2\left[ {1 + \cos \left( {\dfrac{\pi }{2} - \theta } \right)} \right]$
On simplifying this using, the formula ${\sin ^2}x + {\cos ^2}x = 1$ we get
$\Rightarrow$ $2\sqrt {{{\sin }^2}\theta } + 2\left[ {1 + \cos \left( {\dfrac{\pi }{2} - \theta } \right)} \right]$
Now let’s take $2$ commonly out from the above expression.
$\Rightarrow$ $2\left( {\sqrt {{{\sin }^2}\theta } + \left[ {1 + \cos \left( {\dfrac{\pi }{2} - \theta } \right)} \right]} \right)$
On simplifying this expression, we get
$\Rightarrow$ $2\left( {\left| {\sin \theta } \right| + \left[ {1 + \cos \left( {\dfrac{\pi }{2} - \theta } \right)} \right]} \right)$
Using the formula $\cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta$ we get
$\Rightarrow$ $2\left( {\left| {\sin \theta } \right| + \left[ {1 + \sin \theta } \right]} \right)$
We know that the sine function is negative in the range $\pi < \theta < \dfrac{{3\pi }}{2}$ . Applying this we get
$\Rightarrow$ $2\left( { - \sin \theta + \left[ {1 + \sin \theta } \right]} \right)$
Now let us multiply $2$ to the terms inside.
$\Rightarrow$ $- 2\sin \theta + 2 + 2\sin \theta$
Now $- 2\sin \theta$ $\&$ $+ 2\sin \theta$ get canceled. So, we get
$\sqrt {(4{{\sin }^4}\theta + 2{{\sin }^2}2\theta )} + 4{\cos ^2}\left( {\dfrac{\pi }{4} - \dfrac{\theta }{2}} \right)$ $\Rightarrow 2$

So, the correct answer is “Option A”.

Note: To find the value of any complicated function with trigonometric functions we first need to simplify those trigonometric functions using standard formulas of trigonometric functions and their identities. Then we have to check the given range to see whether the function is positive or negative in that range.