$$ \left(1+\tan^2x\right)\sin^2x\text{.} $$Use algebra and the fundamental trigonometric identities.
Your answer should be a number or use a single trigonometric function.


To simplify the expression $ \left(1+\tan^2x\right)\sin^2x $ using algebra and fundamental trigonometric identities, let's follow these steps:

1. **Recall the Pythagorean Identity for $\tan^2x$:**

   \[
   1 + \tan^2x = \sec^2x
   \]

2. **Substitute $ \sec^2x $ into the expression:**

   \[
   \left(1 + \tan^2x\right)\sin^2x = \sec^2x \cdot \sin^2x
   \]

3. **Express $\sec x$ in terms of $\cos x$:**

   \[
   \sec x = \frac{1}{\cos x}
   \]
   
   Therefore,
   
   \[
   \sec^2x = \frac{1}{\cos^2 x}
   \]

4. **Substitute $\frac{1}{\cos^2 x}$ for $\sec^2 x$:**

   \[
   \sec^2x \cdot \sin^2x = \frac{1}{\cos^2 x} \cdot \sin^2x
   \]

5. **Simplify the expression:**

   \[
   \frac{\sin^2 x}{\cos^2 x}
   \]

6. **Recognize the resulting trigonometric identity:**

   \[
   \frac{\sin^2 x}{\cos^2 x} = \tan^2 x
   \]

So, the simplified form of the given expression is:

\[
\tan^2 x
\]

Would you like further details or have any questions?

Here are some related questions that might help you expand your understanding:

1. How do you derive the Pythagorean identity involving $\tan x$ and $\sec x$?
2. What is the relationship between $\sin x$ and $\cos x$ in trigonometric identities?
3. How can you simplify $\sec^2 x$ in terms of other trigonometric functions?
4. What are the other fundamental Pythagorean identities in trigonometry?
5. How does the identity $\tan x = \frac{\sin x}{\cos x}$ help in simplifying trigonometric expressions?
6. Can you use similar steps to simplify $\left(1 + \cot^2 x\right)\cos^2 x$?
7. What are some common mistakes to avoid when using trigonometric identities?
8. How can understanding these identities help in solving more complex trigonometric equations?

**Tip:** Always verify the domain of the trigonometric function you are working with to avoid undefined values, such as when $\cos x = 0$.

Learn how to simplify the trigonometric expression (1 + tan^2 x) sin^2 x step-by-step with algebra and fundamental trigonometric identities. Discover the relationship between secant, cosine, sine, and tangent in trigonometry. Suitable for high school students and anyone interested in advanced trigonometric identities.

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