Can you express 8(sin⁡x)48(sin*x*)4 in terms of cosine and sine using trigonometric identities?

Yes, we can express \( 8(\sin x)^4 \) in terms of both sine and cosine using trigonometric identities.

First, let's use the double-angle identity for sine:

\[ \sin^2 x = \frac{1 - \cos 2x}{2} \]

Now, we want to find \(\sin^4 x\), which can be done by squaring \(\sin^2 x\):

\[ \sin^4 x = \left( \frac{1 - \cos 2x}{2} \right)^2 = \frac{(1 - \cos 2x)^2}{4} \]

Expand the square in the numerator:

\[ (1 - \cos 2x)^2 = 1 - 2\cos 2x + \cos^2 2x \]

Therefore:

\[ \sin^4 x = \frac{1 - 2\cos 2x + \cos^2 2x}{4} \]

Now, substitute this back into the original expression:

\[ 8(\sin x)^4 = 8 \cdot \frac{1 - 2\cos 2x + \cos^2 2x}{4} \]

Simplify by multiplying both the numerator and the denominator by 2:

\[ 8(\sin x)^4 = 2(1 - 2\cos 2x + \cos^2 2x) \]

Distribute the 2:

\[ 8(\sin x)^4 = 2 - 4\cos 2x + 2\cos^2 2x \]

So, the expression \( 8(\sin x)^4 \) in terms of cosine and sine is:

\[ 8(\sin x)^4 = 2 - 4\cos 2x + 2\cos^2 2x \]

Would you like further details or have any specific questions about this transformation?

Here are some relative questions that expand on this information:

1. How would you graph the function \( f(x) = 2 - 4\cos 2x + 2\cos^2 2x \)?
2. What is the derivative of \( 2 - 4\cos 2x + 2\cos^2 2x \) with respect to \( x \)?
3. How does the expression \( 2 - 4\cos 2x + 2\cos^2 2x \) change if we replace \( x \) with \( x/2 \)?
4. What are the critical points of the function \( f(x) = 2 - 4\cos 2x + 2\cos^2 2x \)?
5. Can you express \( 2 - 4\cos 2x + 2\cos^2 2x \) in a simplified form using other trigonometric identities?
6. What is the integral of \( 2 - 4\cos 2x + 2\cos^2 2x \) with respect to \( x \)?
7. How does the expression \( 2 - 4\cos 2x + 2\cos^2 2x \) behave as \( x \) approaches infinity?
8. What is the Taylor series expansion of \( 2 - 4\cos 2x + 2\cos^2 2x \) around \( x = 0 \)?

**Tip:** When dealing with trigonometric identities, always look for opportunities to apply Pythagorean identities, double-angle formulas, and other trigonometric relationships to simplify or transform expressions.

Learn how to express the expression 8(sin⁡x)48(sin*x*)4 in terms of both sine and cosine using trigonometric identities. This detailed solution applies the double-angle identity for sine and simplifies the expression step-by-step. Suitable for high school students studying trigonometry.

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