Question

Evaluate $\left(\sec 2\mathrm{A}+1\right){\sec }^2\mathrm{A}=$

• A. $\sec \mathrm{A}$
• B. $2\sec \mathrm{A}$
• C. $\sec 2\mathrm{A}$
• D. $2\sec 2\mathrm{A}$

Answer 1

The correct option is D

$2\sec 2\mathrm{A}$

Solve the function:

We know, $\sec \mathrm{A}=\frac{1}{\cos A}\cdots \left(1\right)$

With the help of equation $\left(1\right)$, we can rewrite the question as:

$$\begin{gathered} \left(\sec 2\mathrm{A}+1\right){\sec }^2\mathrm{A}=\left(\frac{1}{\cos 2A}+1\right)\frac{1}{{\cos }^{2}A} \\ =\left(\frac{1+\cos 2A}{\cos 2A}\right)\frac{1}{{\cos }^{2}A} \\ =\left(\frac{2{\cos }^{2}A}{\cos 2A}\right)\frac{1}{{\cos }^{2}A}[\because \cos 2A=2{\cos }^{2}A-1] \\ =\frac{2}{\cos 2A} \\ =2\sec 2\mathrm{A}[\because \cos 2A=\frac{1}{\sec 2\mathrm{A}}] \end{gathered}$$

Hence, the correct answer is option (D).