Certainly! Let's solve this trigonometric equation step-by-step. The given equation is:
[tex]\[
(1 - \sin A + \cos A)^2 = 2(1 - \sin A)(1 + \cos A)
\][/tex]
To solve this equation, we need to simplify and verify both sides separately.
### Step 1: Simplify the Left-Hand Side (LHS)
The left-hand side of the equation is:
[tex]\[
(1 - \sin A + \cos A)^2
\][/tex]
We'll expand this expression using the binomial theorem:
[tex]\[
(1 - \sin A + \cos A)^2 = (1 - \sin A + \cos A)(1 - \sin A + \cos A)
\][/tex]
Using the distributive property (also known as FOIL for binomials), we get:
[tex]\[
(1 - \sin A + \cos A)(1 - \sin A + \cos A) = 1 \cdot 1 + 1 \cdot (-\sin A) + 1 \cdot \cos A - \sin A \cdot 1 - \sin A \cdot (-\sin A) - \sin A \cdot \cos A + \cos A \cdot 1 + \cos A \cdot (-\sin A) + \cos A \cdot \cos A
\][/tex]
Combining like terms:
[tex]\[
= 1 - \sin A + \cos A - \sin A + \sin^2 A - \sin A \cos A + \cos A - \sin A \cos A + \cos^2 A
\][/tex]
[tex]\[
= 1 - 2\sin A + \cos A + \cos A + \sin^2 A + \cos^2 A - 2 \sin A \cos A
\][/tex]
Since [tex]\(\sin^2 A + \cos^2 A = 1\)[/tex]:
[tex]\[
= 1 - 2 \sin A + 2 \cos A + 1 - 2 \sin A \cos A
\][/tex]
[tex]\[
= 2 - 2 \sin A + 2 \cos A - 2 \sin A \cos A
\][/tex]
To simplify further:
[tex]\[
= 2 + 2 (\cos A - \sin A - \sin A \cos A)
\][/tex]
This is already simplified.
### Step 2: Simplify the Right-Hand Side (RHS)
The right-hand side of the equation is:
[tex]\[
2(1 - \sin A)(1 + \cos A)
\][/tex]
We'll expand this expression:
[tex]\[
2(1 - \sin A)(1 + \cos A) = 2 \times [(1 \cdot 1) + (1 \cdot \cos A) - (\sin A \cdot 1) - (\sin A \cdot \cos A)]
\][/tex]
Combining like terms:
[tex]\[
= 2(1 + \cos A - \sin A - \sin A \cos A)
\][/tex]
Distribute the 2:
[tex]\[
= 2 \cdot 1 + 2 \cos A - 2 \sin A - 2 \sin A \cos A
\][/tex]
[tex]\[
= 2 + 2 \cos A - 2 \sin A - 2 \sin A \cos A
\][/tex]
### Step 3: Comparing LHS and RHS
Now, both sides have been simplified:
[tex]\[
\text{LHS}: 2 + 2 \cos A - 2 \sin A - 2 \sin A \cos A
\][/tex]
[tex]\[
\text{RHS}: 2 + 2 \cos A - 2 \sin A - 2 \sin A \cos A
\][/tex]
Interestingly, the simplified forms of the left-hand side and right-hand side match exactly. Thus, the equation:
[tex]\[
(1 - \sin A + \cos A)^2 = 2(1 - \sin A)(1 + \cos A)
\][/tex]
is indeed true, as the simplified expressions of both sides are equivalent.
