which function is the same as $y = 3cosleft2left(x+\frac{pi}{2}\right)\right-2$?\n$y = 3sinleft2left(x+\frac{…

Explanation:

Step1: Use the trigonometric identity $\cos(A + \frac{\pi}{2})=-\sin(A)$

Let $A = 2(x+\frac{\pi}{4})$. First, expand the original function $y = 3\cos(2(x+\frac{\pi}{2}))-2=3\cos(2x+\pi)-2$. We know that $\cos(A + \pi)=-\cos(A)$, so $y=- 3\cos(2x)-2$. Also, using the co - function identity $\cos(x+\frac{\pi}{2})=-\sin(x)$. For the function $y = 3\cos(2(x+\frac{\pi}{2}))-2$, we can rewrite it as follows: [ \begin{align*} y&=3\cos(2x+\pi)-2\ &=3\cos\left(2\left(x + \frac{\pi}{4}+\frac{\pi}{4}\right)\right)-2\ &=3\cos\left(2\left(x+\frac{\pi}{4}\right)+\frac{\pi}{2}\right)-2 \end{align*} ] By the identity $\cos(\alpha+\frac{\pi}{2})=-\sin(\alpha)$ where $\alpha = 2(x + \frac{\pi}{4})$, we get $y=-3\sin(2(x+\frac{\pi}{4}))-2$.

Answer:

$y=-3\sin(2(x+\frac{\pi}{4}))-2$ (the second option)