If sin a + sin² a = 1,then cos² a + cos⁴ a=

Given the equation:

\[
\sin a + \sin^2 a = 1
\]

We need to find the value of \( \cos^2 a + \cos^4 a \).

### Step 1: Simplify the given equation

We start by solving for \( \sin a \) from the given equation. Assume \( \sin a = x \). The equation becomes:

\[
x + x^2 = 1
\]

This is a quadratic equation. Rearranging it gives:

\[
x^2 + x - 1 = 0
\]

### Step 2: Solve the quadratic equation

We solve for \( x \) (which is \( \sin a \)) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting \( a = 1 \), \( b = 1 \), and \( c = -1 \):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 4}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{5}}{2}
\]

Thus, \( \sin a = \frac{-1 + \sqrt{5}}{2} \) (since sine cannot be negative for this scenario).

### Step 3: Find \( \cos^2 a \)

We know that:

\[
\sin^2 a + \cos^2 a = 1
\]

Thus:

\[
\cos^2 a = 1 - \sin^2 a
\]

Substituting \( \sin a = \frac{-1 + \sqrt{5}}{2} \):

\[
\cos^2 a = 1 - \left(\frac{-1 + \sqrt{5}}{2}\right)^2
\]

Simplifying:

\[
\cos^2 a = 1 - \frac{(-1 + \sqrt{5})^2}{4}
\]

Expanding the square:

\[
\cos^2 a = 1 - \frac{1 - 2\sqrt{5} + 5}{4}
\]

\[
\cos^2 a = 1 - \frac{6 - 2\sqrt{5}}{4}
\]

\[
\cos^2 a = 1 - \frac{3 - \sqrt{5}}{2}
\]

\[
\cos^2 a = \frac{2 - (3 - \sqrt{5})}{2} = \frac{-1 + \sqrt{5}}{2}
\]

### Step 4: Find \( \cos^2 a + \cos^4 a \)

We now need to calculate \( \cos^4 a \):

\[
\cos^4 a = \left(\cos^2 a\right)^2 = \left(\frac{-1 + \sqrt{5}}{2}\right)^2
\]

Thus:

\[
\cos^4 a = \frac{6 - 2\sqrt{5}}{4} = \frac{3 - \sqrt{5}}{2}
\]

Finally, sum \( \cos^2 a \) and \( \cos^4 a \):

\[
\cos^2 a + \cos^4 a = \frac{-1 + \sqrt{5}}{2} + \frac{3 - \sqrt{5}}{2} = \frac{2}{2} = 1
\]

Therefore, the value of \( \cos^2 a + \cos^4 a \) is \( \boxed{1} \).

---

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the steps to solve quadratic equations?
2. How do you derive trigonometric identities from given equations?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How can you verify trigonometric equations?
5. What are some other methods to solve for \( \sin a \) and \( \cos a \) in trigonometric equations?

**Tip:** Remember that squaring equations can introduce extraneous solutions, so always check back against the original equation.

Learn how to solve the trigonometric equation sin a + sin^2 a = 1 to find the value of cos^2 a + cos^4 a. This problem involves using the quadratic formula and trigonometric identities. Suitable for advanced high school students.

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