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We are given the function \( y = a \cos^2 x + b \sin^2 x \), and we need to show that:

\[
\frac{d^2 y}{dx^2} + 4y = 2(a + b)
\]

### Step 1: First Derivative of \( y \)

First, we take the first derivative of \( y \) with respect to \( x \):

\[
y = a \cos^2 x + b \sin^2 x
\]

Using the chain rule and the derivatives of trigonometric functions:

\[
\frac{d}{dx}(\cos^2 x) = 2\cos x (-\sin x) = -2 \cos x \sin x
\]
\[
\frac{d}{dx}(\sin^2 x) = 2 \sin x \cos x
\]

Thus, the first derivative of \( y \) is:

\[
\frac{dy}{dx} = a(-2 \cos x \sin x) + b(2 \sin x \cos x)
\]
\[
\frac{dy}{dx} = 2 \cos x \sin x (b - a)
\]

### Step 2: Second Derivative of \( y \)

Now, take the second derivative of \( y \):

\[
\frac{d}{dx}(2 \cos x \sin x (b - a)) = (b - a) \frac{d}{dx}(2 \cos x \sin x)
\]

We use the product rule for differentiation:

\[
\frac{d}{dx}(2 \cos x \sin x) = 2 \left( -\sin^2 x + \cos^2 x \right)
\]

Thus, the second derivative becomes:

\[
\frac{d^2 y}{dx^2} = (b - a) 2 (\cos^2 x - \sin^2 x)
\]

### Step 3: Apply the Given Condition

We now substitute this into the equation we need to prove:

\[
\frac{d^2 y}{dx^2} + 4y = 2(a + b)
\]

Substitute \( y = a \cos^2 x + b \sin^2 x \) and \( \frac{d^2 y}{dx^2} = 2(b - a)(\cos^2 x - \sin^2 x) \):

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

Expand the equation:

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

Notice that the terms involving \( \cos^2 x \) and \( \sin^2 x \) cancel out, leaving:

\[
2a + 2b = 2(a + b)
\]

Thus, the equation holds, and the original statement is proven.

Would you like further details or have any questions?

### Related Questions:
1. How would you compute the first derivative of \( \cos^2 x \) from scratch?
2. Why does the product rule apply when differentiating \( 2 \cos x \sin x \)?
3. Can you describe the trigonometric identity used in the simplification?
4. How does the cancellation of terms help in proving the equation?
5. How would the result change if the coefficients \( a \) and \( b \) were functions of \( x \)?

**Tip:** When differentiating trigonometric functions, keep chain rule and product rule in mind to avoid mistakes.

Given that y = a * cos^2(x) + b * sin^2(x), show that d^2y/dx^2 + 4y = 2(a + b).

Explore how to differentiate the trigonometric function y = a cos^2(x) + b sin^2(x) and prove that d^2y/dx^2 + 4y = 2(a + b). Learn the steps involving the product rule, chain rule, and key trigonometric identities.

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