Question

$(\sec \theta + \tan \theta)(1 - \sin \theta)$ is equal to:

• A. $\sec \theta$
• B. $\sin \theta$
• C. $\cosec \theta$
• D. $\cos \theta$

Answer 1

The Correct Option is D

Step 1: Expand the given expression:
We are given the expression \( (\sec \theta + \tan \theta)(1 - \sin \theta) \), and we are tasked with simplifying it.
To simplify this, we can expand the product using the distributive property (also known as the FOIL method for binomials):
\[ (\sec \theta + \tan \theta)(1 - \sin \theta) = \sec \theta (1 - \sin \theta) + \tan \theta (1 - \sin \theta) \] Step 2: Simplify each term separately:
First, simplify \( \sec \theta (1 - \sin \theta) \):
\[ \sec \theta (1 - \sin \theta) = \sec \theta - \sec \theta \sin \theta \] Next, simplify \( \tan \theta (1 - \sin \theta) \):
\[ \tan \theta (1 - \sin \theta) = \tan \theta - \tan \theta \sin \theta \] Now, the expression becomes:
\[ \sec \theta - \sec \theta \sin \theta + \tan \theta - \tan \theta \sin \theta \] Step 3: Use trigonometric identities to further simplify:
We know the following trigonometric identities: - \( \sec \theta = \frac{1}{\cos \theta} \) - \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) Substitute these identities into the expression: \[ \frac{1}{\cos \theta} - \frac{1}{\cos \theta} \sin \theta + \frac{\sin \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta} \sin \theta \] Step 4: Combine like terms:
The expression simplifies to: \[ \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta} \] Notice that the terms \( \frac{\sin \theta}{\cos \theta} \) cancel each other out, leaving us with: \[ \frac{1}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta} \] Factor out \( \frac{1}{\cos \theta} \): \[ \frac{1}{\cos \theta} \left( 1 - \sin^2 \theta \right) \] Step 5: Use the Pythagorean identity:
We know that \( 1 - \sin^2 \theta = \cos^2 \theta \). Therefore, the expression becomes: \[ \frac{1}{\cos \theta} \times \cos^2 \theta = \cos \theta \] Step 6: Conclusion:
The simplified expression is \( \boxed{\cos \theta} \).