There are multiple ways to convert or rewrite $\sin^2 x$ depending on the context of the problem. Below are some common transformations:

1. Using the Pythagorean Identity

From the fundamental Pythagorean identity:
$\sin^2 x + \cos^2 x = 1$ Solving for $\sin^2 x$: $\sin^2 x = 1 - \cos^2 x$ This form is useful when you need to express $\sin^2 x$ in terms of $\cos x$.

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2. Using the Double-Angle Identity of Cosine

From the double-angle identity: $\cos 2x = \cos^2 x - \sin^2 x$ Using $\cos^2 x = 1 - \sin^2 x$, we rewrite: $\cos 2x = 1 - 2\sin^2 x$ Solving for $\sin^2 x$: $\sin^2 x = \frac{1 - \cos 2x}{2}$ This transformation is particularly useful in integration and trigonometric simplifications.

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3. Using the Half-Angle Identity

The half-angle identity provides another way to express $\sin^2 x$: $\sin^2 x = \frac{1 - \cos 2x}{2}$ This is the same as the previous result derived from the double-angle identity.

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4. Using Power Reduction Formula

Since $\sin^2 x$ appears in trigonometric power reduction formulas, the most common transformation is: $\sin^2 x = \frac{1 - \cos 2x}{2}$ which is useful in integrals involving even powers of sine.

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5. In Terms of Exponentials (Euler's Formula)

Using Euler’s formula: $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ Squaring both sides: $\sin^2 x = \left(\frac{e^{ix} - e^{-ix}}{2i}\right)^2$ Expanding: $\sin^2 x = \frac{- (e^{2ix} - 2 + e^{-2ix})}{4}$ Simplifying: $\sin^2 x = \frac{1 - e^{2ix} - e^{-2ix}}{4}$ Using $e^{2ix} + e^{-2ix} = 2\cos 2x$, we get: $\sin^2 x = \frac{1 - \cos 2x}{2}$ This again leads to the well-known power-reduction formula.

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Summary of Conversions:

$\sin^2 x = 1 - \cos^2 x$ $\sin^2 x = \frac{1 - \cos 2x}{2}$ $\sin^2 x = \frac{1 - e^{2ix} - e^{-2ix}}{4}$

Would you like a step-by-step derivation for any of these?

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Related Questions:

1. How can we express $\cos^2 x$ in different forms?
2. What is the integral of $\sin^2 x$?
3. How does the power reduction formula help in trigonometric integrals?
4. How do we use Euler’s formula to derive trigonometric identities?
5. What are the applications of the double-angle formulas?

Tip:

When solving trigonometric integrals, the power reduction identity is particularly useful to convert squared sine and cosine terms into forms that are easier to integrate.