verify the identity. (simplify at each step.) sin(5π/2 + x) = cos(x) sin(5π/2 + x) = sin(5π/2)( ) +…

Explanation:

Step1: Apply the sum - formula for sine

The sum - formula for sine is $\sin(A + B)=\sin(A)\cos(B)+\cos(A)\sin(B)$. Here $A = \frac{5\pi}{2}$ and $B=x$. So $\sin(\frac{5\pi}{2}+x)=\sin(\frac{5\pi}{2})\cos(x)+\cos(\frac{5\pi}{2})\sin(x)$.

Step2: Evaluate trigonometric values

We know that $\sin(\frac{5\pi}{2})=\sin(2\pi+\frac{\pi}{2})$. Since $\sin(2k\pi +\alpha)=\sin(\alpha)$ for any integer $k$ and real - number $\alpha$, then $\sin(\frac{5\pi}{2})=\sin(\frac{\pi}{2}) = 1$. Also, $\cos(\frac{5\pi}{2})=\cos(2\pi+\frac{\pi}{2})$. Since $\cos(2k\pi+\alpha)=\cos(\alpha)$ for any integer $k$ and real - number $\alpha$, then $\cos(\frac{5\pi}{2})=\cos(\frac{\pi}{2}) = 0$.

Step3: Substitute values

Substituting $\sin(\frac{5\pi}{2}) = 1$ and $\cos(\frac{5\pi}{2}) = 0$ into $\sin(\frac{5\pi}{2})\cos(x)+\cos(\frac{5\pi}{2})\sin(x)$, we get $(1)\cos(x)+(0)\sin(x)=\cos(x)$.

Answer:

The identity $\sin(\frac{5\pi}{2}+x)=\cos(x)$ is verified.