Certainly! Let's solve the given expression step-by-step:
We are given the expression:
[tex]\[ (\sin A + \cos A)^2 - (\sin A - \cos A)^2 \][/tex]
Our goal is to show that this simplifies to:
[tex]\[ 4 \sin A \cos A \][/tex]
First, we'll expand each square term:
1. Expand [tex]\((\sin A + \cos A)^2\)[/tex]:
[tex]\[
(\sin A + \cos A)^2 = (\sin A + \cos A)(\sin A + \cos A)
\][/tex]
Using the distributive property:
[tex]\[
= \sin^2 A + \sin A \cos A + \cos A \sin A + \cos^2 A
\][/tex]
[tex]\[
= \sin^2 A + 2 \sin A \cos A + \cos^2 A
\][/tex]
2. Expand [tex]\((\sin A - \cos A)^2\)[/tex]:
[tex]\[
(\sin A - \cos A)^2 = (\sin A - \cos A)(\sin A - \cos A)
\][/tex]
Using the distributive property:
[tex]\[
= \sin^2 A - \sin A \cos A - \cos A \sin A + \cos^2 A
\][/tex]
[tex]\[
= \sin^2 A - 2 \sin A \cos A + \cos^2 A
\][/tex]
Next, we subtract these two expanded expressions:
[tex]\[
(\sin^2 A + 2 \sin A \cos A + \cos^2 A) - (\sin^2 A - 2 \sin A \cos A + \cos^2 A)
\][/tex]
Combine like terms:
[tex]\[
(\sin^2 A + 2 \sin A \cos A + \cos^2 A) - \sin^2 A + 2 \sin A \cos A - \cos^2 A
\][/tex]
[tex]\[
= \sin^2 A + 2 \sin A \cos A + \cos^2 A - \sin^2 A - \cos^2 A + 2 \sin A \cos A
\][/tex]
Notice that [tex]\(\sin^2 A\)[/tex] and [tex]\(-\sin^2 A\)[/tex] cancel out, likewise for [tex]\(\cos^2 A\)[/tex] and [tex]\(-\cos^2 A\)[/tex]:
[tex]\[
= 2 \sin A \cos A + 2 \sin A \cos A
\][/tex]
[tex]\[
= 4 \sin A \cos A
\][/tex]
Therefore, we have shown that:
[tex]\[
(\sin A + \cos A)^2 - (\sin A - \cos A)^2 = 4 \sin A \cos A
\][/tex]
This confirms the given result.
