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.