2.12 Use 2.11 to write down two other expressions representing $$ \cos 2 \alpha $$ HINT USE THE Identity $$ \sin ^{2} heta+\cos ^{2} heta=1 $$

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2.12 Use 2.11 to write down two other expressions representing $$ \cos 2 \alpha $$ HINT USE THE Identity $$ \sin ^{2} 	heta+\cos ^{2} 	heta=1 $$

UpStudy AI · Accepted Answer

1. Start with the double‐angle identity:
$$
\cos 2\alpha = \cos^{2}\alpha - \sin^{2}\alpha
$$

2. Express the above in terms of only $$\cos \alpha$$. Use the identity:
$$
\sin^{2}\alpha + \cos^{2}\alpha = 1 \quad \Rightarrow \quad \sin^{2}\alpha = 1 - \cos^{2}\alpha
$$
Substitute into the double‐angle formula:
$$
\cos 2\alpha = \cos^{2}\alpha - \left(1-\cos^{2}\alpha\right) = 2\cos^{2}\alpha - 1
$$

3. Express the double‐angle formula in terms of only $$\sin \alpha$$. Again use the identity:
$$
\cos^{2}\alpha = 1 - \sin^{2}\alpha
$$
Substitute into the double‐angle formula:
$$
\cos 2\alpha = \left(1-\sin^{2}\alpha\right) - \sin^{2}\alpha = 1- 2\sin^{2}\alpha
$$

Thus, the two other expressions representing $$\cos 2\alpha$$ are:
$$
\cos 2\alpha = 2\cos^{2}\alpha - 1 \quad \text{and} \quad \cos 2\alpha = 1 - 2\sin^{2}\alpha.
$$
$$ \cos 2\alpha = 2\cos^{2}\alpha - 1 \quad \text{and} \quad \cos 2\alpha = 1 - 2\sin^{2}\alpha. $$

2.12 Use 2.11 to write down two other expressions representing

HINT USE THE Identity

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2.12 Use 2.11 to write down two other expressions representing HINT USE THE Identity