Tangent (π- α) × cos (π/2 +α) > 0. the answer is that cos α < 0. can someone explain why?

Supplementary angles have opposite tangents

tan(π - a) = - tan a

Adding  π/2 turns a cosine into a negated sine.

cos(π/2 + a) = - sin a

These are easily shown by the various sum angle formulas; I won't bother here.

The numerator is never negative so if the whole thing is positive so is the denominator:

That's the opposite conclusion to the question but I think it's right.  Let's check a=π/4.

tan(π - π/4) = tan(3π/4) = -1

cos(π/2 + π/4) = cos(3π/4) = -1/√2

The product is positive.

So is cos(π/4).

I'm right.

it's equivalent, so we can say that its equal.

let's call the denominator x and the numerador is 18 less then denominator, x - 18.

multiply

2.x = 5.(x - 18)

2x = 5x - 90

90 = 5x - 2x

90 = 3x

x = 90/3

x = 30

so: