Question

The value of $(\sec A + \tan A)(1 - \sin A)$ will be:

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

Hint:

In trigonometric simplifications, always convert into $\sin A$ and $\cos A$ first. Then use $1 - \sin^2 A = \cos^2 A$ for quick reduction.

Answer 1

The Correct Option is D

Step 1: Write in terms of $\sin A$ and $\cos A$
\[ \sec A = \frac{1}{\cos A}, \tan A = \frac{\sin A}{\cos A} \] So, \[ \sec A + \tan A = \frac{1 + \sin A}{\cos A} \]

Step 2: Multiply with $(1 - \sin A)$
\[ (\sec A + \tan A)(1 - \sin A) = \frac{1 + \sin A}{\cos A} \times (1 - \sin A) \] \[ = \frac{(1 + \sin A)(1 - \sin A)}{\cos A} \] \[ = \frac{1 - \sin^2 A}{\cos A} \]

Step 3: Simplify using identity
Since $1 - \sin^2 A = \cos^2 A$: \[ = \frac{\cos^2 A}{\cos A} = \cos A \]

Step 4: Conclusion
Thus, \[ (\sec A + \tan A)(1 - \sin A) = \cos A \] The correct answer is option (D).