Tan(x+pi) + 2sin(x+pi)=0

can you prove this and make the left side equal the right?

This is not an identity but an equation. You're supposed to find the values of for which this is true. It is *not* true for all values of .

To see why:

has period , which means .

The angle sum identity for says that

So

By definition of ,

and so

which is only 0 if either (which only happens for certain values of ) or (which also only happens for certain values of ). It's these values of you want to find.

whenever is a multiple of , i.e. for any integer .

If , then is true for and . Then to account for all other possible values, we add a multiple of , so that or for integers .

if the eqn of the parabola is f(x) = (x-4)(x+2), we need to mult. this out to find the coordinates of the vertex.

then f(x) = x^2 + 2x - 4x - 8, and f(x) = x^2 - 2x - 8.

one way in which you can find the vertex is to calculate x = -b / (2a).   here, that comes to

        )

x = = 1.   f(1) = 1^2 - 2(1) - 8 = - 9.   vertex is at (1, -9).

        2(1) 

find another point on the curve:   let x=0, for which y=-8.   then (0, -8) is on the curve.   plot the vertex and this point and draw a smooth parabola thru both.

hope this