Answer :

GinnyAnswer
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.

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