Question

If x is acute angle, and tan x=3 , evaluate cos x−sin x
_ __________
4 cos x+sin x

Answer 1

Use / as your division sign, enclose numerators and denominators in brackets.

e.g. cos x−sin x
_ __________
4 cos x+sin x

could mean: cosx/4 - sinx/(cosx + sinx)

I will assume that is what you meant.

given tanx = 3 = 3/1 = y/x , so make a sketch of your triangle
and x^2 + y^2 = r^2
1^2 + 3^2 = r^2
r = √10
since x is acute, sinx = 3/√10 , and cosx = 1/√10

cosx/4 - sinx/(cosx + sinx)
= 1/(4√10) - (3/√10) / (4√10)
= 1/(4√10) - 3/4
= <b>(1 - 3√10)/(4√10)</b>

or (√10 - 30)/40 after rationalizing.