To simplify the expression [tex]\(\left(1 + \tan^2 \alpha\right)(1 - \sin \alpha)(1 + \sin \alpha)\)[/tex], let's proceed step-by-step.
1. Recognize the identity for tangent squared:
[tex]\[
1 + \tan^2 \alpha
\][/tex]
This is a well-known trigonometric identity, which can be expressed as:
[tex]\[
1 + \tan^2 \alpha = \sec^2 \alpha
\][/tex]
2. Factorize the sine terms:
[tex]\[
(1 - \sin \alpha)(1 + \sin \alpha)
\][/tex]
This is a difference of squares. Recall the identity:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
So,
[tex]\[
(1 - \sin \alpha)(1 + \sin \alpha) = 1^2 - (\sin \alpha)^2 = 1 - \sin^2 \alpha
\][/tex]
3. Use the Pythagorean identity:
[tex]\[
1 - \sin^2 \alpha
\][/tex]
This can be simplified using the Pythagorean identity:
[tex]\[
\sin^2 \alpha + \cos^2 \alpha = 1
\][/tex]
Thus,
[tex]\[
1 - \sin^2 \alpha = \cos^2 \alpha
\][/tex]
4. Combine the results:
Substitute the simplified forms into the original expression:
[tex]\[
\left(1 + \tan^2 \alpha\right)(1 - \sin \alpha)(1 + \sin \alpha) = \sec^2 \alpha \cdot \cos^2 \alpha
\][/tex]
5. Simplify the product:
Recall the definition of secant:
[tex]\[
\sec \alpha = \frac{1}{\cos \alpha}
\][/tex]
Therefore:
[tex]\[
\sec^2 \alpha = \frac{1}{\cos^2 \alpha}
\][/tex]
Substitute back:
[tex]\[
\sec^2 \alpha \cdot \cos^2 \alpha = \frac{1}{\cos^2 \alpha} \cdot \cos^2 \alpha
\][/tex]
6. Final simplification:
Since [tex]\(\frac{\cos^2 \alpha}{\cos^2 \alpha} = 1\)[/tex], the simplified expression is:
[tex]\[
1
\][/tex]
Thus, the expression [tex]\(\left(1 + \tan^2 \alpha\right)(1 - \sin \alpha)(1 + \sin \alpha)\)[/tex] simplifies to:
[tex]\[
1
\][/tex]
