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 \( (\sec \theta + \tan \theta)(1 - \sin \theta) \). Apply the distributive property:
\[(\sec \theta + \tan \theta)(1 - \sin \theta) = \sec \theta (1 - \sin \theta) + \tan \theta (1 - \sin \theta)\]Step 2: Simplify each term:
Simplify the first term: \( \sec \theta (1 - \sin \theta) = \sec \theta - \sec \theta \sin \theta \).
Simplify the second term: \( \tan \theta (1 - \sin \theta) = \tan \theta - \tan \theta \sin \theta \).
Combining these, the expression is now:
\[\sec \theta - \sec \theta \sin \theta + \tan \theta - \tan \theta \sin \theta\]Step 3: Apply trigonometric identities:
Use \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute these 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 terms:
Rewrite the expression with common denominators:
\[\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta}\]The middle two terms cancel. The expression becomes:
\[\frac{1}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta}\]Factor out \( \frac{1}{\cos \theta} \):
\[\frac{1}{\cos \theta} \left( 1 - \sin^2 \theta \right)\]Step 5: Use the Pythagorean identity:
Since \( 1 - \sin^2 \theta = \cos^2 \theta \), substitute this in:
\[\frac{1}{\cos \theta} \times \cos^2 \theta = \cos \theta\]Step 6: Conclusion:
The simplified expression is \( \boxed{\cos \theta} \).