Questions: sin^2(x) sec^2(x) - sin^2(x)

Transcript text: $\sin ^{2}(x) \sec ^{2}(x)-\sin ^{2}(x)$

Solution

Solution Steps

To simplify the given expression, we can use trigonometric identities. Recall that $\sec(x) = \frac{1}{\cos(x)}$. Therefore, $\sec^2(x) = \frac{1}{\cos^2(x)}$. We can substitute this into the expression and simplify.

Solution Approach

Substitute $\sec^2(x)$ with $\frac{1}{\cos^2(x)}$.
Simplify the expression by combining like terms.

Step 1: Substitute $\sec^2(x)$

We start with the expression: \[ \sin^2(x) \sec^2(x) - \sin^2(x) \] Using the identity $\sec^2(x) = \frac{1}{\cos^2(x)}$, we can rewrite the expression as: \[ \sin^2(x) \cdot \frac{1}{\cos^2(x)} - \sin^2(x) \]

Step 2: Combine Like Terms

Next, we factor out $\sin^2(x)$ from the expression: \[ \sin^2(x) \left(\frac{1}{\cos^2(x)} - 1\right) \] We can simplify the term inside the parentheses: \[ \frac{1}{\cos^2(x)} - 1 = \frac{1 - \cos^2(x)}{\cos^2(x)} = \frac{\sin^2(x)}{\cos^2(x)} \] Thus, the expression becomes: \[ \sin^2(x) \cdot \frac{\sin^2(x)}{\cos^2(x)} = \frac{\sin^4(x)}{\cos^2(x)} \]

Step 3: Use the Identity for $\tan^2(x)$

Recognizing that $\frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x)$, we can rewrite the expression as: \[ \sin^2(x) \tan^2(x) \]