Verify that the following equation is an identity.

To verify the identity, start with the more complicated side and transform it to look like the other side. Ch

Ask by Washington Bowers. in the United States

Jan 21,2025

Question

Verify that the following equation is an identity.

To verify the identity, start with the more complicated side and transform it to look like the other side. Ch

Ask by Washington Bowers. in the United States

Jan 21,2025

UpStudy AI · Accepted Answer

Sure, let's verify the trigonometric identity:

$$
(\cos 2x + \sin 2x)^{2} = 1 + \sin 4x
$$

**Step 1: Expand the Left-Hand Side (LHS)**

Start with the more complicated side, which is the left-hand side (LHS):

$$
(\cos 2x + \sin 2x)^{2}
$$

Use the algebraic identity $$(a + b)^2 = a^2 + 2ab + b^2$$ to expand the expression:

$$
(\cos 2x + \sin 2x)^{2} = \cos^2 2x + 2 \cos 2x \sin 2x + \sin^2 2x
$$

**Step 2: Simplify Using Trigonometric Identities**

1. **Combine $$\cos^2 2x$$ and $$\sin^2 2x$$:**

Recall the Pythagorean identity:

$$
   \cos^2 \theta + \sin^2 \theta = 1
   $$

Applying this to $$\theta = 2x$$:

$$
   \cos^2 2x + \sin^2 2x = 1
   $$

2. **Simplify the Middle Term:**

The middle term is $$2 \cos 2x \sin 2x$$. Use the double-angle identity for sine:

$$
   2 \sin \theta \cos \theta = \sin 2\theta
   $$

Applying this to $$\theta = 2x$$:

$$
   2 \cos 2x \sin 2x = \sin 4x
   $$

**Step 3: Combine the Simplified Terms**

Now, substitute the simplified terms back into the expanded expression:

$$
\cos^2 2x + 2 \cos 2x \sin 2x + \sin^2 2x = 1 + \sin 4x
$$

**Conclusion**

Thus, we have:

$$
(\cos 2x + \sin 2x)^{2} = 1 + \sin 4x
$$

This confirms that the given equation is indeed an identity.
$$ (\cos 2x + \sin 2x)^{2} = 1 + \sin 4x $$ This equation is an identity.