Problem 33 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

Using the co-function identity \(\sin (\frac{\pi}{2} + y) = \cos (\pi - y)\), it can be verified that \(\sin \left(x+\frac{\pi}{2}\right)=\cos x\).

Step 1: Apply the co-function identity

The co-function identity states that \(\sin (\frac{\pi}{2} - y) = \cos y\). To use this identity, we need to rearrange the given equation so that it fits this form. Our given expression doesn't have a subtraction, but rather an addition \(\left(x+\frac{\pi}{2}\right)\). This might be a point of confusion. However, we can think of this in terms of angle rotation. Remember that sine and cosine are periodic with period \(2\pi\). Therefore, we can write the identity as \(\sin (\frac{\pi}{2} + y) = \cos (\pi - y)\). The angle \(\pi - y\) rotates an angle \(y\) in the reversed direction, that is, from \(\pi\) to \(y\). Using this adjusted form of the co-function identity, we prove that \(\sin \left(x+\frac{\pi}{2}\right)=\cos x\).

Step 2: Verify the identity

The given equation is \(\sin \left(x+\frac{\pi}{2}\right)=\cos x\). We can think of \(\frac{\pi}{2} + x\) as \(y\) in our adjusted co-function identity. Therefore, we can write \(\sin (\frac{\pi}{2} + x) = \cos (\pi - x)\). Now this fits precisely the form of our adjusted co-function identity, hence verifying that \(\sin \left(x+\frac{\pi}{2}\right)=\cos x\).