Question

Which of the following is equivalent to cosine of 2 times x over the quantity cosine of x end quantity question mark

Answer 1

Given,

$\frac{\cos 2x}{cosx}$

By using the below formula,

$\cos 2x=\cos ^2x-\sin ^2x$

Now, substitute this formula.

$\begin{gathered} \frac{\cos^2x-\sin^2x}{\cos x}=\frac{\cos^2x}{\cos x}-\frac{\sin ^2x}{\cos x} \\ =\cos x-\sin x\frac{\sin x}{\cos x} \\ =\cos x-\sin x\tan x \end{gathered}$

Now, use the formula.

$\cos 2x=2\cos ^2x-1$

Substitute this formula.

$\begin{gathered} \frac{2\cos^2x-1}{\cos x}=\frac{2\cos^2x}{\cos x}-\frac{1}{\cos x} \\ =2\cos x-\sec x \end{gathered}$

Therefore, the given options (i) and (iii) are correct.