Question

Simplest form of $\frac{2}{\sqrt{2+\left(\sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)}\right)}}$ is

• A. $\sec \left(\frac{x}{2}\right)$
• B. $\sec \left(x\right)$
• C. $\csc \left(x\right)$
• D. $1$

Answer 1

The correct option is A

$\sec \left(\frac{x}{2}\right)$

Explanation for the correct option:

Simplifying the given expression:

$\frac{2}{\sqrt{2+\left(\sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)}\right)}}...\left(i\right)$

Now Solving,

$$\begin{gathered} \Rightarrow 2+2\cos 4x \\ \Rightarrow 2\left(1+\cos 2\left(2x\right)\right) \\ \Rightarrow 2\left(2{\cos }^2\left(2x\right)\right) \\ \Rightarrow 4{\cos }^2\left(2x\right) \end{gathered}$$

Therefore,

$\sqrt{4{\cos }^2\left(2x\right)}=2\cos 2x$

Now,

$$\begin{gathered} \Rightarrow \sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)} \\ \Rightarrow \sqrt{2+2\cos x} \\ \Rightarrow \sqrt{2(1+\cos 2x)} \\ \Rightarrow \sqrt{2\left(2{\cos }^{2}x\right)} \\ \Rightarrow 2cosx \end{gathered}$$

And

$$\begin{gathered} \sqrt{2+\left(\sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)}\right)} \\ \Rightarrow \sqrt{2+2\cos x} \\ \Rightarrow \sqrt{2(1+\cos x)} \\ \Rightarrow \sqrt{2\left(2{\cos }^2\frac{x}{2}\right)} \\ =2\cos \left(\frac{x}{2}\right)...\left(ii\right) \end{gathered}$$

From $\left(i\right)$ and $\left(ii\right)$

$$\begin{gathered} \frac{2}{\sqrt{2+\left(\sqrt{2+\left(\sqrt{\left(2+2\cos 4x\right)}\right)}\right)}} \\ \Rightarrow \frac{2}{2\cos \left({\displaystyle \frac{x}{2}}\right)} \\ \Rightarrow \sec \left(\frac{x}{2}\right) \end{gathered}$$

Therefore, the correct answer is option (A).