Certainly! Let's simplify the given expression step by step:
[tex]\[
\sec^2\left(\frac{\pi}{2} - x\right)\left[\sin^2(x) - \sin^4(x)\right]
\][/tex]
### Step 1: Simplify the secant term
We know the secant function [tex]\(\sec\)[/tex] is the reciprocal of the cosine function [tex]\(\cos\)[/tex]:
[tex]\[
\sec\theta = \frac{1}{\cos\theta}
\][/tex]
Given [tex]\(\theta = \frac{\pi}{2} - x\)[/tex]:
[tex]\[
\sec\left(\frac{\pi}{2} - x\right) = \frac{1}{\cos\left(\frac{\pi}{2} - x\right)}
\][/tex]
We can use the co-function identity for cosine:
[tex]\[
\cos\left(\frac{\pi}{2} - x\right) = \sin(x)
\][/tex]
Thus:
[tex]\[
\sec\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin(x)}
\][/tex]
Then squared:
[tex]\[
\sec^2\left(\frac{\pi}{2} - x\right) = \left(\frac{1}{\sin(x)}\right)^2 = \frac{1}{\sin^2(x)}
\][/tex]
### Step 2: Substitute and Simplify
Now substitute [tex]\(\sec^2\left(\frac{\pi}{2} - x\right)\)[/tex] back into the original expression:
[tex]\[
\frac{1}{\sin^2(x)} \left[\sin^2(x) - \sin^4(x)\right]
\][/tex]
Distribute the term [tex]\(\frac{1}{\sin^2(x)}\)[/tex] over the terms in the brackets:
[tex]\[
\frac{\sin^2(x)}{\sin^2(x)} - \frac{\sin^4(x)}{\sin^2(x)}
\][/tex]
This simplifies to:
[tex]\[
1 - \sin^2(x)
\][/tex]
### Step 3: Use Pythagorean Identity
We know from the Pythagorean identity that:
[tex]\[
1 - \sin^2(x) = \cos^2(x)
\][/tex]
### Conclusion
Thus, the simplified form of the given expression is:
[tex]\[
\cos^2(x)
\][/tex]
This completes the simplification of the given expression. The final, simplified result is:
[tex]\[
\cos^2(x)
\][/tex]
