If cos x +cos²x =1 approve sin⁴ x+sin² x=1

We are given the equation:

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

We need to prove that:

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

### Step 1: Work with the given equation

Start by isolating \( \cos^2(x) \) in the given equation:

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

### Step 2: Use the Pythagorean identity

We know the fundamental trigonometric identity:

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

Substitute \( \cos^2(x) = 1 - \sin^2(x) \) into the equation from Step 1:

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

This simplifies to:

\[
\sin^2(x) = \cos(x)
\]

### Step 3: Work with the target expression

We are now aiming to prove:

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

From Step 2, we know that \( \sin^2(x) = \cos(x) \). Substitute this into the target expression:

\[
(\sin^2(x))^2 + \sin^2(x) = 1
\]

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

This is exactly the given equation, \( \cos(x) + \cos^2(x) = 1 \).

Thus, the original equation holds, and we have proved that:

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

### Conclusion:

We have successfully proven that \( \sin^4(x) + \sin^2(x) = 1 \) follows from \( \cos(x) + \cos^2(x) = 1 \).

Would you like more details on any step, or have any further questions?

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Here are 5 related questions for deeper understanding:

1. What is the Pythagorean identity in trigonometry, and how is it derived?
2. How does the Pythagorean identity help solve trigonometric equations?
3. Can you prove other trigonometric identities using this method?
4. How do sine and cosine relate geometrically on the unit circle?
5. What are the other fundamental trigonometric identities?

**Tip:** Always check the Pythagorean identity first when dealing with expressions involving both sine and cosine. It simplifies many problems!

If cos x + cos²x = 1, prove sin⁴ x + sin² x = 1

Learn how to prove that sin⁴(x) + sin²(x) = 1, given the equation cos(x) + cos²(x) = 1. This step-by-step solution covers key trigonometric concepts and the use of the Pythagorean identity. Suitable for students in Grades 10-12.

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