We are tasked with proving the trigonometric identity:

\[
1 - \cos^2x = \sin x \cos x \tan x
\]

### Step-by-step proof:

#### Step 1: Recall the Pythagorean identity
One of the fundamental trigonometric identities is:

\[
\sin^2x + \cos^2x = 1
\]

From this, we can express \( \sin^2x \) as:

\[
\sin^2x = 1 - \cos^2x
\]

Thus, the left-hand side (LHS) of the equation \( 1 - \cos^2x \) is equivalent to \( \sin^2x \). Therefore, we rewrite the LHS as:

\[
\sin^2x
\]

So the equation becomes:

\[
\sin^2x = \sin x \cos x \tan x
\]

#### Step 2: Simplify the right-hand side
Now, let's simplify the right-hand side (RHS) \( \sin x \cos x \tan x \). Recall the definition of the tangent function:

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

Substituting this into the RHS:

\[
\sin x \cos x \tan x = \sin x \cos x \cdot \frac{\sin x}{\cos x}
\]

The \( \cos x \) terms cancel out:

\[
\sin x \cdot \sin x = \sin^2x
\]

Thus, the RHS simplifies to:

\[
\sin^2x
\]

#### Step 3: Conclusion
Since both the LHS and the RHS are equal to \( \sin^2x \), we have:

\[
1 - \cos^2x = \sin x \cos x \tan x
\]

Thus, the identity is proven.

---

Would you like more details or have any further questions on this?

Here are five related questions to expand your understanding:
1. How is the Pythagorean identity derived in trigonometry?
2. Can you prove \( \tan^2x + 1 = \sec^2x \)?
3. How can we prove other trigonometric identities using fundamental identities?
4. What is the geometric interpretation of the sine and cosine functions?
5. How does the identity \( \sin^2x + \cos^2x = 1 \) relate to the unit circle?

**Tip:** Always keep basic trigonometric identities in mind when solving or simplifying expressions. They are the building blocks for more complex proofs.

Step-by-step solution to proving the trigonometric identity 1 - cos^2x = sinxcosxtanx. This proof involves using the Pythagorean identity and basic trigonometric formulas like sin^2x + cos^2x = 1. Perfect for high school students in Grades 10-12.

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