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Simplify the expression:
[tex]\sec^2\left(\frac{\pi}{2} - x\right)\left[\sin^2(x) - \sin^4(x)\right][/tex]

Sagot :

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]
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