9sin(a-9П)-16cos(5П/2+а)

To simplify the expression \(9\sin(a - 9\pi) - 16\cos\left(\frac{5\pi}{2} + a\right)\), we start by using properties of trigonometric functions.

1. Simplifying \( \sin(a - 9\pi) \): The sine function is periodic with a period of \( 2\pi \). Thus, we can reduce \( 9\pi \) modulo \( 2\pi \): \[ 9\pi \mod 2\pi = \pi \] Therefore, \[ \sin(a - 9\pi) = \sin(a - \pi) = -\sin(a) \]

2. Simplifying \( \cos\left(\frac{5\pi}{2} + a\right) \): The cosine function is also periodic with a period of \( 2\pi \). We can simplify \( \frac{5\pi}{2} \): \[ \frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2} \] Thus, we have: \[ \cos\left(\frac{5\pi}{2} + a\right) = \cos\left(\frac{\pi}{2} + a\right) = -\sin(a) \]

Now substituting these simplifications back into the original expression: \[ 9\sin(a - 9\pi) - 16\cos\left(\frac{5\pi}{2} + a\right) = 9(-\sin(a)) - 16(-\sin(a)) \] \[ = -9\sin(a) + 16\sin(a) = (16 - 9)\sin(a) = 7\sin(a) \]

Thus, the simplified expression is: \[ \boxed{7\sin(a)} \]