To simplify the expression [tex]\(\sin\left(\frac{\pi}{2} - x\right) \sec\left(\frac{\pi}{2} - x\right)\)[/tex], we can use some trigonometric identities. Let's proceed step-by-step:
1. Co-function Identity for Sine and Cosine:
The co-function identity states that [tex]\(\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\)[/tex]. This is because the sine of an angle is equal to the cosine of its complement.
So, [tex]\(\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\)[/tex].
2. Express Secant in Terms of Cosine:
Recall that [tex]\(\sec\theta = \frac{1}{\cos\theta}\)[/tex]. This means [tex]\(\sec\left(\frac{\pi}{2} - x\right) = \frac{1}{\cos\left(\frac{\pi}{2} - x\right)}\)[/tex].
3. Co-function Identity for Secant:
Another co-function identity states that [tex]\(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\)[/tex]. Therefore, [tex]\(\sec\left(\frac{\pi}{2} - x\right) = \frac{1}{\sin(x)}\)[/tex].
4. Combine the Identities:
Substituting these identities back into the original expression, we have:
[tex]\[
\sin\left(\frac{\pi}{2} - x\right) \sec\left(\frac{\pi}{2} - x\right) = \cos(x) \cdot \frac{1}{\sin(x)}
\][/tex]
5. Further Simplify the Expression:
However, notice we should apply the identity directly without confusing cases. Instead, from previous identities:
- [tex]\(\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\)[/tex]
- [tex]\(\sec\left(\frac{\pi}{2} - x\right) = \csc(x)\)[/tex] from co-function itself, simplifying directly.
6. Final Simplification:
- [tex]\(\cos(x) \left( \frac{1}{\cos(x)} \right)\)[/tex]
Given [tex]\(\cos(x)\)[/tex] and its reciprocal results:
[tex]\[
\cos(x) * \sec(x) = \cos(x) \left( \frac{1}{\cos(x)} \right) = 1
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{1}
\][/tex]
