To solve the equation [tex]\(3.\)[/tex] [tex]\(2 \cos^2 A - 1 = \cos^2 A - \sin^2 A\)[/tex], let's go through the steps methodically.
### Step 1: Recall the Pythagorean Identity
We start with the fundamental Pythagorean identity:
[tex]\[ \sin^2 A + \cos^2 A = 1 \][/tex]
From this identity, we can solve for [tex]\(\sin^2 A\)[/tex]:
[tex]\[ \sin^2 A = 1 - \cos^2 A \][/tex]
### Step 2: Substitute [tex]\(\sin^2 A\)[/tex] in the Given Equation
The given equation is:
[tex]\[ 2 \cos^2 A - 1 = \cos^2 A - \sin^2 A \][/tex]
Substitute [tex]\(\sin^2 A = 1 - \cos^2 A\)[/tex] into the equation on the right-hand side:
[tex]\[ \cos^2 A - (1 - \cos^2 A) \][/tex]
### Step 3: Simplify the Equation
Simplify the expression inside the parenthesis:
[tex]\[ \cos^2 A - 1 + \cos^2 A \][/tex]
[tex]\[ = \cos^2 A + \cos^2 A - 1 \][/tex]
[tex]\[ = 2 \cos^2 A - 1 \][/tex]
### Step 4: Verify Both Sides of the Equation
After the substitution and simplification, both sides of the equation become:
[tex]\[ 2 \cos^2 A - 1 = 2 \cos^2 A - 1 \][/tex]
### Conclusion
We see that the left-hand side is exactly equal to the right-hand side:
[tex]\[ 2 \cos^2 A - 1 = 2 \cos^2 A - 1 \][/tex]
This demonstrates that the given equation holds true for all values of [tex]\(A\)[/tex]. Thus, it is an identity.
Therefore, the equation [tex]\(2 \cos^2 A - 1 = \cos^2 A - \sin^2 A\)[/tex] is indeed an identity and holds true for all values of [tex]\( A \)[/tex].
