$$ 2 \sin ^ { 2 } ( 45 ^ { \circ } - A ) $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ 2 \sin^2(45^\circ - A) $$, we can use the double angle identity for sine. The double angle identity states that:

$$
\sin^2 x = \frac{1 - \cos(2x)}{2}
$$

Applying this identity to our expression, we have:

$$
\sin^2(45^\circ - A) = \frac{1 - \cos(2(45^\circ - A))}{2}
$$

Now, substituting this back into the original expression:

$$
2 \sin^2(45^\circ - A) = 2 \cdot \frac{1 - \cos(2(45^\circ - A))}{2}
$$

The 2s cancel out:

$$
= 1 - \cos(2(45^\circ - A))
$$

Next, we simplify $$ \cos(2(45^\circ - A)) $$:

$$
\cos(2(45^\circ - A)) = \cos(90^\circ - 2A) = \sin(2A)
$$

Thus, we can rewrite our expression as:

$$
1 - \sin(2A)
$$

So, the simplified form of $$ 2 \sin^2(45^\circ - A) $$ is:

$$
\boxed{1 - \sin(2A)}
$$
$$ 1 - \sin(2A) $$

\( 2 \sin ^ { 2 } ( 45 ^ { \circ } - A ) \)