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: Expressing the given expression:
We are asked to simplify the expression \((\sec \theta + \tan \theta)(1 - \sin \theta)\).
We will use trigonometric identities to simplify the given expression.

Step 2: Expanding the expression:
We can start by expanding the given expression: \[ (\sec \theta + \tan \theta)(1 - \sin \theta) = \sec \theta (1 - \sin \theta) + \tan \theta (1 - \sin \theta) \] Now, let's simplify each term.

Step 3: Simplifying the first term:
For \(\sec \theta (1 - \sin \theta)\), we can write \(\sec \theta = \frac{1}{\cos \theta}\), so: \[ \sec \theta (1 - \sin \theta) = \frac{1}{\cos \theta} (1 - \sin \theta) \] This becomes: \[ \frac{1 - \sin \theta}{\cos \theta} \]

Step 4: Simplifying the second term:
For \(\tan \theta (1 - \sin \theta)\), we can write \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so: \[ \tan \theta (1 - \sin \theta) = \frac{\sin \theta}{\cos \theta} (1 - \sin \theta) \] This becomes: \[ \frac{\sin \theta (1 - \sin \theta)}{\cos \theta} \]

Step 5: Combining the two terms:
Now, we can combine both terms: \[ \frac{1 - \sin \theta}{\cos \theta} + \frac{\sin \theta (1 - \sin \theta)}{\cos \theta} \] Since both terms have a common denominator of \(\cos \theta\), we can combine them into a single fraction: \[ \frac{(1 - \sin \theta) + \sin \theta (1 - \sin \theta)}{\cos \theta} \] Simplifying the numerator: \[ (1 - \sin \theta) + \sin \theta (1 - \sin \theta) = 1 - \sin \theta + \sin \theta - \sin^2 \theta \] This simplifies to: \[ 1 - \sin^2 \theta \] Since \(1 - \sin^2 \theta = \cos^2 \theta\), we now have: \[ \frac{\cos^2 \theta}{\cos \theta} \] Simplifying further: \[ \cos \theta \]

Step 6: Conclusion:
The expression \((\sec \theta + \tan \theta)(1 - \sin \theta)\) simplifies to \(\cos \theta\).