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: Objective:
The task is to simplify the expression \((\sec \theta + \tan \theta)(1 - \sin \theta)\) using trigonometric identities.

Step 2: Expansion:
Expand the expression:
\[ (\sec \theta + \tan \theta)(1 - \sin \theta) = \sec \theta (1 - \sin \theta) + \tan \theta (1 - \sin \theta) \]

Step 3: Simplify First Term:
Rewrite \(\sec \theta\) as \(\frac{1}{\cos \theta}\):
\[ \sec \theta (1 - \sin \theta) = \frac{1}{\cos \theta} (1 - \sin \theta) = \frac{1 - \sin \theta}{\cos \theta} \]

Step 4: Simplify Second Term:
Rewrite \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\):
\[ \tan \theta (1 - \sin \theta) = \frac{\sin \theta}{\cos \theta} (1 - \sin \theta) = \frac{\sin \theta (1 - \sin \theta)}{\cos \theta} \]

Step 5: Combination and Simplification:
Combine the simplified terms:
\[ \frac{1 - \sin \theta}{\cos \theta} + \frac{\sin \theta (1 - \sin \theta)}{\cos \theta} = \frac{(1 - \sin \theta) + \sin \theta (1 - \sin \theta)}{\cos \theta} \]
Simplify the numerator:
\[ (1 - \sin \theta) + \sin \theta - \sin^2 \theta = 1 - \sin^2 \theta \]
Using the identity \(1 - \sin^2 \theta = \cos^2 \theta\):
\[ \frac{\cos^2 \theta}{\cos \theta} \]
Further simplification yields:
\[ \cos \theta \]

Step 6: Result:
The simplified form of \((\sec \theta + \tan \theta)(1 - \sin \theta)\) is \(\cos \theta\).