Question

If we have an equation as $8\cos 2\theta +8\sec 2\theta =65,0 < \theta < \dfrac{\pi }{2}$ then the value of $4\cos 4\theta $ is equal to:
(1) $\dfrac{-33}{8}$
(2) $\dfrac{-31}{8}$
(3) $\dfrac{-31}{32}$
(4) $\dfrac{-33}{32}$

Answer 1

Hint: Here in this question we have been asked to find the value of $4\cos 4\theta $ given that $8\cos 2\theta +8\sec 2\theta =65,0 < \theta < \dfrac{\pi }{2}$ . To answer this question we will convert $\sec 2\theta $ into $\cos 2\theta $ and then simplify the quadratic expression that we get and find the value of $\cos 2\theta $ and use it to derive the required result.

Complete step-by-step solution:
Now considering from the question we have been asked to find the value of $4\cos 4\theta $ given that $8\cos 2\theta +8\sec 2\theta =65,0 < \theta < \dfrac{\pi }{2}$ .
Now by using $\sec x=\dfrac{1}{\cos x}$in the given expression, we will have $ 8\cos 2\theta +8\left( \dfrac{1}{\cos 2\theta } \right)=65$ .
Now by further simplifying this expression we will have $ 8{{\cos }^{2}}2\theta +8=65\cos 2\theta $.
By further simplifying this expression we will have
$\begin{align}
  & \Rightarrow 8{{\cos }^{2}}2\theta -64\cos 2\theta -\cos 2\theta +8=0 \\
 & \Rightarrow \left( 8\cos 2\theta -1 \right)\left( \cos 2\theta -8 \right)=0 \\
\end{align}$ .
Now we know that the value of $\cos x$ always lies between -1 and 1.
Hence now we will have $\cos 2\theta =\dfrac{1}{8}$ .
From the basic concepts of trigonometry, we know that $\cos 2x=2{{\cos }^{2}}x-1$ .
Hence now we can say that
$\begin{align}
  & \cos 4\theta =2{{\cos }^{2}}2\theta -1 \\
 & \Rightarrow 2{{\left( \dfrac{1}{8} \right)}^{2}}-1=\dfrac{1}{32}-1 \\
 & \Rightarrow \dfrac{-31}{32} \\
\end{align}$ .
Therefore we can conclude that when it is given that $8\cos 2\theta +8\sec 2\theta =65,0 < \theta < \dfrac{\pi }{2}$ then we will have the value of $4\cos 4\theta $ as $\dfrac{-31}{8}$ .
Hence we will mark the option “2” as correct.

Note: While answering questions of this type we should be sure with the formula that we are going to use in the process of simplification in between the steps. This question seems to be a complex one but when we start applying the appropriate it gets solved within a short span of time. The formula for $\cos 2x$ can also be given as $1-2{{\sin }^{2}}x={{\cos }^{2}}x-{{\sin }^{2}}x$ .