Question

$\left[\frac{\sin 2A}{1+\cos 2A}\right]\left[\frac{\cos 2A}{1+\cos A}\right]=$

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

Answer 1

The correct option is A

$\tan \left(\frac{A}{2}\right)$

Explanation for the correct answer:

Simplifying the given expression:

$\left[\frac{\sin 2A}{1+\cos 2A}\right]\left[\frac{\cos 2A}{1+\cos A}\right]$

Using the formula$\cos 2x=2{\cos }^{2}x–1and\sin 2x=2\sin x\cos x$

$$\begin{gathered} \left(\frac{2\sin A\cos A}{2{\cos }^{2}A}\right)\left(\frac{\cos A}{1+\cos A}\right) \\ =\frac{\sin A}{1+\cos A} \end{gathered}$$

Using the formulas $\sin x=2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)and\cos x=2{\cos }^2\left(\frac{x}{2}\right)–1$

$$\begin{gathered} =\frac{2\sin \left({\displaystyle \frac{A}{2}}\right)\cos \left({\displaystyle \frac{A}{2}}\right)}{2{\cos }^2\left({\displaystyle \frac{A}{2}}\right)} \\ =\frac{\sin \left({\displaystyle \frac{A}{2}}\right)}{\cos \left({\displaystyle \frac{A}{2}}\right)} \\ =\tan \left(\frac{A}{2}\right) \end{gathered}$$

Therefore, the correct answer is option (A).