Sure! Let's go through the steps to derive the cofunction identity \(\sin \left(\frac{\pi}{2} - x \right) = \cos(x)\).
### Step-by-Step Derivation
1. Starting Expression:
[tex]\[
\sin \left( \frac{\pi}{2} - x \right)
\][/tex]
2. Use the angle addition formula for sine:
[tex]\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\][/tex]
Applying this formula where \(A = \frac{\pi}{2}\) and \(B = x\), we get:
[tex]\[
\sin \left( \frac{\pi}{2} - x \right) = \sin \left( \frac{\pi}{2} \right) \cos(x) - \cos \left( \frac{\pi}{2} \right) \sin(x)
\][/tex]
3. Evaluate \(\sin \left( \frac{\pi}{2} \right)\) and \(\cos \left( \frac{\pi}{2} \right)\):
From the unit circle or standard trigonometric values:
[tex]\[
\sin \left( \frac{\pi}{2} \right) = 1
\][/tex]
[tex]\[
\cos \left( \frac{\pi}{2} \right) = 0
\][/tex]
4. Substitute these values back into the expression:
[tex]\[
\sin \left( \frac{\pi}{2} - x \right) = 1 \cdot \cos(x) - 0 \cdot \sin(x)
\][/tex]
5. Simplify the expression:
[tex]\[
\sin \left( \frac{\pi}{2} - x \right) = \cos(x)
\][/tex]
Thus, we have derived the cofunction identity:
[tex]\[
\sin \left( \frac{\pi}{2} - x \right) = \cos(x)
\][/tex]
This completes the derivation of the cofunction identity.
