ART 5: DEDUCTION AND APPLICATION: o not use a calculator when answering Part 5 . 1 Apply the compound angle expansion $$ \cos (x+y)=\cos x \cos y-\sin x \sin y $$ to $$ \cos 2 x $$ and simpl your answer: $$ \qquad $$ $$ \cos 2 x=\cos (x+x) $$ $$ = $$ $$ = $$ $$ \qquad $$ 2 Complete the square identity: $$ \sin ^{2} x+\cos ^{2} x=\ldots \ldots \ldots $$ 3 Use the identity in 5.2 and rewrite $$ \cos ^{2} x-\sin ^{2} x $$ in terms of $$ \sin x $$ only: $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ 4 Use the identity in 5.2 and rewrite $$ \cos ^{2} x-\sin ^{2} x $$ in terms of $$ \cos x $$ only:

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1. We start with the compound angle identity:
$$
\cos (x+y)=\cos x\, \cos y-\sin x\, \sin y.
$$
For $$ \cos 2x $$, substitute $$ y=x $$:
$$
\cos 2x=\cos (x+x)=\cos x\, \cos x-\sin x\, \sin x=\cos^2 x-\sin^2 x.
$$

2. The complete square (Pythagorean) identity is:
$$
\sin^2 x+\cos^2 x=1.
$$

3. To write $$ \cos^2 x-\sin^2 x $$ in terms of $$ \sin x $$ only, express $$ \cos^2 x $$ using the identity:
$$
\cos^2 x=1-\sin^2 x.
$$
Substitute into the expression:
$$
\cos^2 x-\sin^2 x=(1-\sin^2 x)-\sin^2 x=1-2\sin^2 x.
$$

4. To express $$ \cos^2 x-\sin^2 x $$ in terms of $$ \cos x $$ only, solve for $$ \sin^2 x $$ from the identity:
$$
\sin^2 x=1-\cos^2 x.
$$
Substitute into the expression:
$$
\cos^2 x-\sin^2 x=\cos^2 x-(1-\cos^2 x)=2\cos^2 x-1.
$$
1. $$ \cos 2x = \cos^2 x - \sin^2 x $$ 2. $$ \sin^2 x + \cos^2 x = 1 $$ 3. $$ \cos^2 x - \sin^2 x = 1 - 2\sin^2 x $$ 4. $$ \cos^2 x - \sin^2 x = 2\cos^2 x - 1 $$

ART 5: DEDUCTION AND APPLICATION:
o not use a calculator when answering Part 5 .
1 Apply the compound angle expansion to and simpl your answer:

2 Complete the square identity:

3 Use the identity in 5.2 and rewrite in terms of only:

4 Use the identity in 5.2 and rewrite in terms of only:

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ART 5: DEDUCTION AND APPLICATION: o not use a calculator when answering Part 5 . 1 Apply the compound angle expansion to and simpl your answer: 2 Complete the square identity: 3 Use the identity in 5.2 and rewrite in terms of only: 4 Use the identity in 5.2 and rewrite in terms of only: