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

Question

UpStudy AI · Accepted Answer

We start with the expression:

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

Since $$\frac{1}{1}=1$$, the expression is:

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

**Step 1. Rewrite the denominator using a trigonometric identity**

Recall the double-angle identity:

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

Thus, the square of the denominator is:

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

Now the expression becomes:

$$
1 + \frac{\sin^2 4x}{4\cos^4 2x}
$$

**Step 2. Rewrite the numerator using a double-angle identity for sine**

We use the identity for the sine of a double angle:

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

Squaring both sides gives:

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

Substitute this back into the expression:

$$
1 + \frac{4\sin^2 2x \cos^2 2x}{4\cos^4 2x}
$$

**Step 3. Simplify the expression**

Cancel the common factor of 4 in the numerator and the denominator:

$$
1 + \frac{\sin^2 2x \cos^2 2x}{\cos^4 2x}
$$

Simplify the fraction by canceling $$\cos^2 2x$$:

$$
1 + \frac{\sin^2 2x}{\cos^2 2x} = 1 + \tan^2 2x
$$

**Step 4. Use the Pythagorean identity**

Recall the Pythagorean identity:

$$
1 + \tan^2 \theta = \sec^2 \theta
$$

With $$\theta = 2x$$, the expression becomes:

$$
\sec^2 2x
$$

Thus, the final simplified result is:

$$
\boxed{\sec^2 2x}
$$
The simplified expression is $$ \sec^2 2x $$.

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