If x is an acute angle and tan x =3/4 evaluate cos x - sin x/cos x + sin x

Answer 1

To find the value of the expression, let's first find the values of sin(x) and cos(x).

Given that tan(x) = 3/4, we can draw a right triangle with the opposite side measuring 3 and the adjacent side measuring 4. Using the Pythagorean theorem, we can find the length of the hypotenuse.

Hypotenuse² = Opposite² + Adjacent²
Hypotenuse² = 3² + 4²
Hypotenuse² = 9 + 16
Hypotenuse² = 25

Taking the square root of both sides, we have:
Hypotenuse = √25
Hypotenuse = 5

Since sin(x) = Opposite/Hypotenuse, we have:
sin(x) = 3/5

Similarly, cos(x) = Adjacent/Hypotenuse:
cos(x) = 4/5

Now, substitute the values of sin(x) and cos(x) into the expression:

(cos(x) - sin(x))/(cos(x) + sin(x))
= (4/5 - 3/5)/(4/5 + 3/5)
= (1/5)/(7/5)
= (1/5) * (5/7)
= 1/7

Therefore, cos(x) - sin(x)/cos(x) + sin(x) = 1/7.