If is a n acute angle and tanx=3 4 evaluate cosx-sinx cosx+sinx

To evaluate cos(x) - sin(x) and cos(x) + sin(x), we need to find the values of cos(x) and sin(x) first. Since tan(x) = 3/4, we can use the identity:

tan(x) = sin(x) / cos(x).

Rearranging the equation, we have:

sin(x) = tan(x) * cos(x).

Substituting the given value of tan(x) = 3/4, we have:

sin(x) = (3/4) * cos(x).

We can square both sides of this equation using the identity sin^2(x) + cos^2(x) = 1:

(sin(x))^2 = ((3/4) * cos(x))^2.

Expanding both sides of the equation, we get:

sin^2(x) = (9/16) * cos^2(x).

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute the expression for sin^2(x) to solve for cos^2(x):

(9/16) * cos^2(x) + cos^2(x) = 1.

Combining like terms, we have:

(9/16 + 1) * cos^2(x) = 1.

(25/16) * cos^2(x) = 1.

Dividing both sides by (25/16), we find:

cos^2(x) = 16/25.

Taking the square root of both sides, we obtain:

cos(x) = ±4/5.

Now, we can substitute the value of cos(x) into the equation sin(x) = (3/4) * cos(x) to find sin(x):

sin(x) = (3/4) * (±4/5).

sin(x) = ±3/5.

Therefore, cos(x) - sin(x) = (4/5) - (3/5) = 1/5, and cos(x) + sin(x) = (4/5) + (3/5) = 7/5.

Hence, we have evaluated cos(x) - sin(x) as 1/5, and cos(x) + sin(x) as 7/5.