Question

Choose the correct option. Justify your choice.$(\sec A+\tan A)(1-\sin A)=?$

• A. $\sec A$
• B. $\sin A$
• C. $\csc A$
• D. $\cos A$

Answer 1

The correct option is D

$\cos A$

We know that $\sec A$ is equivalent to $\frac{1}{\cos A}$ and $\tan A$ is equivalent to $\frac{\sin A}{\cos A}$.

Similarly,${\sin }^{2}A+{\cos }^{2}A=1$

Now solve $(\sec A+\tan A)(1-\sin A)$:

$\begin{array}{l}\Rightarrow (\frac{1}{\cos A}+\frac{\sin A}{\cos A})(1-\sin A)\\ \Rightarrow \left(\frac{1+\sin A}{\cos A}\right)(1-\sin A)\\ \Rightarrow \frac{1-{\sin }^{2}A}{\cos A}\\ \Rightarrow \frac{{\cos }^{2}A}{\cos A}\\ \Rightarrow \cos A\end{array}$

Hence, the correct option is (D).