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

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ \cos^2(45^\circ - A) - \sin^2(45^\circ - A) $$, we can use the identity for the difference of squares:

$$
\cos^2 x - \sin^2 x = \cos(2x)
$$

In our case, let $$ x = 45^\circ - A $$. Then we can rewrite the expression as:

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

Now, simplifying $$ 2(45^\circ - A) $$:

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

Thus, we have:

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

Using the co-function identity $$ \cos(90^\circ - x) = \sin x $$, we can further simplify:

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

Therefore, the final result is:

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

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