To solve the equation [tex]\(\sin^2(A) - \cos^2(A) = 2\sin^2(A) - 1\)[/tex], we start by simplifying both sides of the equation step-by-step.
1. Rewrite the equation:
[tex]\[
\sin^2(A) - \cos^2(A) = 2\sin^2(A) - 1
\][/tex]
2. Recognize the trigonometric identity for the left-hand side:
Recall the double angle identity for cosine, which states that:
[tex]\[
\cos(2A) = \cos^2(A) - \sin^2(A)
\][/tex]
Therefore, [tex]\(\sin^2(A) - \cos^2(A)\)[/tex] can be rewritten using this identity:
[tex]\[
\sin^2(A) - \cos^2(A) = -(\cos^2(A) - \sin^2(A)) = -\cos(2A)
\][/tex]
3. Expression for the Right-hand Side:
The right-hand side of the equation is already simplified:
[tex]\[
2\sin^2(A) - 1
\][/tex]
4. Equalize both expressions:
Now, we need to determine whether both sides of the equation become equal when simplified. So check:
[tex]\[
-\cos(2A) \stackrel{?}{=} 2\sin^2(A) - 1
\][/tex]
5. Simplify the right-hand side using another trigonometric identity:
Recall that [tex]\(\sin^2(A)\)[/tex] can be expressed in terms of cosine using the Pythagorean identity:
[tex]\[
\sin^2(A) = \frac{1 - \cos(2A)}{2}
\][/tex]
Substitute this into the right-hand side:
[tex]\[
2\sin^2(A) - 1 = 2 \left( \frac{1 - \cos(2A)}{2} \right) - 1
\][/tex]
Simplify the expression:
[tex]\[
2 \left( \frac{1 - \cos(2A)}{2} \right) - 1 = 1 - \cos(2A) - 1 = -\cos(2A)
\][/tex]
6. Compare the simplified expressions:
Both sides simplify to:
[tex]\[
-\cos(2A)
\][/tex]
Therefore, the original equation:
[tex]\[
\sin^2(A) - \cos^2(A) = 2\sin^2(A) - 1
\][/tex]
holds true because the simplified left-hand side and the right-hand side are equivalent to [tex]\(-\cos(2A)\)[/tex].
Thus, the simplified forms [tex]\( \sin^2(A) - \cos^2(A) \)[/tex] and [tex]\( 2\sin^2(A) - 1 \)[/tex] are indeed equal, confirming the identity. The final equation [tex]\(-\cos(2A) = -\cos(2A)\)[/tex] is true always.
