The problem in the image asks to multiply and simplify the expression:

$(1 - \sin \theta)(1 + \sin \theta)$

Solution:

This expression is in the form of a difference of squares, which is a well-known algebraic identity:

$(a - b)(a + b) = a^2 - b^2$

Here, $a = 1$ and $b = \sin \theta$. Applying the identity:

$(1 - \sin \theta)(1 + \sin \theta) = 1^2 - (\sin \theta)^2$

Simplify this to:

$1 - \sin^2 \theta$

Now, recall the Pythagorean identity for trigonometry:

$\sin^2 \theta + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \sin^2 \theta$

Thus:

$1 - \sin^2 \theta = \cos^2 \theta$

Final Answer:

$\cos^2 \theta$

Would you like any further details or have any questions about this problem?

Related Questions:

1. What is the difference of squares formula and how is it used in algebra?
2. How can we derive other Pythagorean identities in trigonometry?
3. What are some applications of the Pythagorean identity in solving trigonometric equations?
4. How can we prove the Pythagorean identity using the unit circle?
5. What is the geometric interpretation of $\cos^2 \theta$ and $\sin^2 \theta$ on the unit circle?

Tip: Whenever you see a product like $(a - b)(a + b)$, think of the difference of squares identity to simplify quickly!