Find the exact value of tan (x-y) if sinx=8/17 and cosy=3/5

To solve the the question we proceed as follows:

From trigonometric laws

(cos x) ^2 + (sin x) ^2=1

(cos y) ^2 + (sin y) ^2=1

sin (x-y) = sin x sin y-sin y cos x

cos (x-y) = cos x cos y + sin x sin y

si x=8/17

cos x=sqrt (1 - (sin x) ^2) = sqrt (1-64/289) = sqrt (225/289) = 15/17

cos y=3/5

sin x = sqrt (1 - (cos x) ^2) = sqrt (1-9/25) = sqrt (16/25) = 4/5

thus

tan (x-y) = [sin (x-y) ]/[cos (x-y) ]

=[sin x cos y-sin y cos x]/[cos x cos y+sin x sin y]

plugging in the values we obtain:

[8/17 * 3/5-4/5*15/7]/[15/17*3/5+8/17*4/5]

simplifying

[24/85-60/85]/[45/85+32/85]

=-36/77