Final answer:
To find the exact value of tan(x-y), we need to know the values of x and y. Given that sinx=8/17 and cosy=3/5, we can find the values of siny and cosy using the trigonometric identities. Using the given values, we can find sin(x-y) and then substitute it into the equation for tan(x-y).
Step-by-step explanation:
To find the exact value of tan(x-y), we need to know the values of x and y. Given that sinx=8/17 and cosy=3/5, we can find the values of siny and cosy using the trigonometric identities. Since sinx=8/17, we can let the opposite side be 8 and the hypotenuse be 17. Using the Pythagorean Theorem, we can find the adjacent side to be sqrt(17^2 - 8^2) = 15. Therefore, cosy = adjacent/hypotenuse = 15/17 = 3/5. Now that we have the values of sinx and cosy, we can find the value of sin(x-y) using the difference formula for sine: sin(x-y) = sinx*cosy - cosx*siny. Plugging in the given values, we get sin(x-y) = (8/17)*(3/5) - (15/17)*(siny). To find siny, we can use the Pythagorean identity: sin^2(y) + cos^2(y) = 1. Plugging in the values of cosy, we get sin^2(y) + (3/5)^2 = 1. Solving for sin(y), we get sin(y) = sqrt(1-(3/5)^2) = sqrt(1-9/25) = sqrt(16/25) = 4/5. Substituting this value into the equation for sin(x-y), we get sin(x-y) = (8/17)*(3/5) - (15/17)*(4/5).