Certainly! Let's verify the identity step-by-step:
1.) [tex]\((1 + \sin x)(1 - \sin x) = \cos^2 x\)[/tex]
### Step 1: Expand the left-hand side (LHS) of the equation
First, let's expand the left-hand side:
[tex]\[
(1 + \sin x)(1 - \sin x)
\][/tex]
This can be expanded using the difference of squares formula. The difference of squares formula states:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
In this case, let [tex]\(a = 1\)[/tex] and [tex]\(b = \sin x\)[/tex]. Therefore, we have:
[tex]\[
(1 + \sin x)(1 - \sin x) = 1^2 - (\sin x)^2
\][/tex]
Simplifying further, we get:
[tex]\[
1 - \sin^2 x
\][/tex]
### Step 2: Use the Pythagorean identity to simplify the expression
Next, we use the Pythagorean identity for trigonometric functions:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Rearranging this identity to express [tex]\(\cos^2 x\)[/tex] in terms of [tex]\(\sin^2 x\)[/tex], we get:
[tex]\[
\cos^2 x = 1 - \sin^2 x
\][/tex]
### Step 3: Compare the expressions
From the result in Step 1, we have:
[tex]\[
1 - \sin^2 x
\][/tex]
And from the Pythagorean identity, we have:
[tex]\[
\cos^2 x = 1 - \sin^2 x
\][/tex]
Therefore, we can conclude that:
[tex]\[
(1 + \sin x)(1 - \sin x) = \cos^2 x
\][/tex]
### Verification
By our step-by-step derivation and the Pythagorean identity, we have confirmed that:
[tex]\[
(1 - \sin x)(1 + \sin x) = \cos^2 x
\][/tex]
Thus, the given identity is verified to be true.
