\frac{1}{\sin ^2\left(x\right)}+\frac{1}{\cos ^2\left(x\right)}=\frac{1}{\sin ^2\left(x\right)-\sin ^4\left(x\right)}

We are given the trigonometric equation:

\[
\frac{1}{\sin ^2(x)} + \frac{1}{\cos ^2(x)} = \frac{1}{\sin ^2(x) - \sin ^4(x)}
\]

Let's verify whether this equation holds true by simplifying both sides.

### Left-hand side (LHS):

\[
\frac{1}{\sin ^2(x)} + \frac{1}{\cos ^2(x)}
\]
We need to combine the two terms by finding a common denominator. The common denominator between \(\sin^2(x)\) and \(\cos^2(x)\) is \(\sin^2(x) \cos^2(x)\).

Thus, we can write:
\[
\frac{1}{\sin ^2(x)} + \frac{1}{\cos ^2(x)} = \frac{\cos^2(x)}{\sin^2(x)\cos^2(x)} + \frac{\sin^2(x)}{\sin^2(x)\cos^2(x)}
\]
This simplifies to:
\[
\frac{\cos^2(x) + \sin^2(x)}{\sin^2(x)\cos^2(x)}
\]
Using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\), we have:
\[
\frac{1}{\sin^2(x)\cos^2(x)}
\]
Thus, the LHS becomes:
\[
\frac{1}{\sin^2(x)\cos^2(x)}
\]

### Right-hand side (RHS):

\[
\frac{1}{\sin^2(x) - \sin^4(x)}
\]
Factor the denominator:
\[
\sin^2(x) - \sin^4(x) = \sin^2(x)(1 - \sin^2(x))
\]
Using the Pythagorean identity again, \(1 - \sin^2(x) = \cos^2(x)\), so:
\[
\sin^2(x)(1 - \sin^2(x)) = \sin^2(x)\cos^2(x)
\]
Thus, the RHS becomes:
\[
\frac{1}{\sin^2(x)\cos^2(x)}
\]

### Conclusion:
Both sides are equal:
\[
\frac{1}{\sin^2(x)\cos^2(x)} = \frac{1}{\sin^2(x)\cos^2(x)}
\]
Hence, the given equation is **true**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the Pythagorean identity help in trigonometric simplifications?
2. Can you solve a similar equation involving \(\tan(x)\) and \(\sec(x)\)?
3. How would the equation change if \(\sin(x)\) were replaced by \(\cos(x)\)?
4. Can you generalize this result for other trigonometric functions?
5. How can this identity be used to solve more complex trigonometric equations?

**Tip**: Always try to simplify trigonometric identities using basic identities like the Pythagorean identity (\(\sin^2(x) + \cos^2(x) = 1\))—it can simplify many expressions.

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

Explore the simplification of the trigonometric equation \frac{1}{\sin ^2(x)} + \frac{1}{\cos ^2(x)} = \frac{1}{\sin ^2(x) - \sin ^4(x)}. This problem demonstrates the use of the Pythagorean identity and other key trigonometric formulas. Suitable for high school students in Grades 10-12.

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