Question

If $8\cos 2\theta +8sec2\theta =65,0<\theta <\frac{\pi }{2},$ then the value of $4\cos 4\theta$ is equal to

• A. $\frac{33}{31}$
• B. $\frac{31}{8}$
• C. $\frac{33}{8}$
• D. $-\frac{31}{8}$

Answer 1

The correct option is D

$-\frac{31}{8}$

Explanation for the correct option:

Step 1. Find the value of $4\cos 4\theta$:

Given, $8\cos 2\theta +8sec2\theta =65,0<\theta <\frac{\pi }{2},$

$\Rightarrow$ $8\cos 2\theta +\frac{8}{\cos 2\theta }=65$

$\Rightarrow$ $8{\cos }^{2}2\theta +8=65\cos 2\theta$

$\Rightarrow$ $8{\cos }^{2}2\theta -65\cos 2\theta +8=0$

$\Rightarrow$ $8{\cos }^{2}2\theta -\cos 2\theta -64\cos 2\theta +8=0$

$\Rightarrow$$\cos 2\theta (8\cos 2\theta -1)-8(8\cos 2\theta -1)=0$

$\Rightarrow$ $(8\cos 2\theta -1)(\cos 2\theta -8)=0$

$\Rightarrow$ $\cos 2\theta =\frac{1}{8}$

Step 2. Put the value of $\cos 2\theta$ in given expression, we get

$\therefore 4\cos 4\theta =4\left[\cos 2\left(2\theta \right)\right]$

$$\begin{gathered} =4(2{\cos }^{2}2\theta -1) \\ =4\left[\left(2\times \frac{1}{64}\right)-1\right] \\ =4\left(-\frac{31}{32}\right) \\ =\mathbf{}\mathbf{-}\frac{\mathbf{31}}{\mathbf{8}} \end{gathered}$$

Hence, Option ‘(D)’ is Correct.