https://file.aiquickdraw.com/mathbot/file/1730875383164irkqr1mr.jpg

To prove the identity \( 4 \sin^4(\theta) = 1 - 2 \cos(2\theta) + \cos^2(2\theta) \), let's go through it step-by-step.

### Step 1: Expanding \( \sin^4(\theta) \) in terms of \( \cos(2\theta) \)

We know that:
\[
\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}.
\]
Therefore,
\[
\sin^4(\theta) = \left(\sin^2(\theta)\right)^2 = \left(\frac{1 - \cos(2\theta)}{2}\right)^2 = \frac{(1 - \cos(2\theta))^2}{4}.
\]

### Step 2: Expanding \( (1 - \cos(2\theta))^2 \)

Now, expand \( (1 - \cos(2\theta))^2 \):
\[
(1 - \cos(2\theta))^2 = 1 - 2\cos(2\theta) + \cos^2(2\theta).
\]
Thus,
\[
\sin^4(\theta) = \frac{1 - 2\cos(2\theta) + \cos^2(2\theta)}{4}.
\]

### Step 3: Multiply both sides by 4

Now, multiply both sides of the equation by 4:
\[
4 \sin^4(\theta) = 1 - 2\cos(2\theta) + \cos^2(2\theta).
\]

### Conclusion

We have shown that:
\[
4 \sin^4(\theta) = 1 - 2 \cos(2\theta) + \cos^2(2\theta),
\]
which completes the proof.

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

Here are some related questions to explore:
1. What are the identities for powers of \(\sin(\theta)\) and \(\cos(\theta)\) in terms of double angles?
2. How can similar techniques be used to prove identities involving higher powers of \(\sin\) or \(\cos\)?
3. What are other methods to prove trigonometric identities?
4. How does expanding powers of trigonometric functions help in simplifying complex trigonometric expressions?
5. Can this method of proof be applied to expressions involving products of different trigonometric functions?

**Tip:** When proving trigonometric identities, converting terms to the same function (like using only sine or cosine) often simplifies the process.

Prove the identity: 4sin^4(θ) = 1 - 2cos(2θ) + cos^2(2θ)

Explore a step-by-step solution for proving the trigonometric identity 4sin^4(θ) = 1 - 2cos(2θ) + cos^2(2θ) using double-angle and power reduction identities. Suitable for high school trigonometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation