Problem 35 - Precalculus Essentials

Question

Verify each identity. $$\cos \left(x-\frac{\pi}{2}\right)=\sin x$$

Step-by-Step Solution

Answer

Verified that \(\cos \left(x-\frac{\pi}{2}\right)=\sin x\) is indeed a valid identity.

Step 1: Analyze the left side of the given identity

Given the identity \(\cos \left(x-\frac{\pi}{2}\right)=\sin x\), start by focusing on the left side of the equation. This expression involves a cosine function, specifically \(\cos\) of some quantity.

Step 2: Use the co-function identity

The co-function identity can be applied here. According to this identity, for all values of x in the domain, \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\). It's important to note that the subtraction of two angles equals the negative of the subtraction in reverse order, meaning \( cos\left(x - \frac{\pi}{2}\right) = cos\left(-\left(\frac{\pi}{2} - x\right)\right)\).

Step 3: Apply the co-function identity

Now apply the co-function identity to the expression on the left side, substituting \(\cos\left(x - \frac{\pi}{2}\right)\) with \(\sin\left(\frac{\pi}{2} - x\right)\).

Step 4: Simplifying the expression

Since any number subtracted from itself results in zero, \(\sin\left(\frac{\pi}{2} - x\right)\) simplifies to \(\sin x\).

Step 5: Verify the identity

Finally, check to see whether the transformed left side is the same as the right side. As the transformed left side is \(\sin x\) and the right side is also \(\sin x\), this verifies that the given trigonometric identity holds true.