What is the maximum integer value of x, where x<200, that satisfies the following inequalities: sin(90+180x/100)<0 and tan(180−180x/100)<0 ?

To find the maximum integer value of x that satisfies the given inequalities, we need to solve each inequality separately.

Let's start with the first inequality, sin(90+180x/100) < 0.

1. Convert degrees to radians: 90 + 180x/100 = (π/2) + (πx/100)
2. Simplify the inequality: sin((π/2) + (πx/100)) < 0
3. Determine the range of x that satisfies this inequality: Since sin is negative in the third and fourth quadrants (-π/2 to π/2), we need to find the values of x that fall within this range.
-π/2 < (π/2) + (πx/100) < π/2 (We exclude π/2 from the range since sin is 0 at π/2)
Subtracting π/2 from all parts of the inequality,
-π/2 - π/2 < πx/100 < π/2 - π/2
-π < πx/100 < 0

Multiplying all parts of the inequality by 100/π (which is positive),
-100 < x < 0

So, the range of x that satisfies the first inequality is -100 < x < 0.

Now, let's move on to the second inequality, tan(180 - 180x/100) < 0.

1. Convert degrees to radians: 180 - 180x/100 = π - (πx/100)
2. Simplify the inequality: tan(π - (πx/100)) < 0
3. Determine the range of x that satisfies this inequality: Since tan is negative in the second and fourth quadrants (-π to 0 and 0 to π), we need to find the values of x that fall within these ranges.
-π < π - (πx/100) < 0 (We exclude 0 from the range since tan is undefined at 0)
Subtracting π from all parts of the inequality,
-2π < - πx/100 < -π

Multiplying all parts of the inequality by 100/π (which is positive),
-200 < x < 100

So, the range of x that satisfies the second inequality is -200 < x < 100.

Now, we need to find the maximum integer value of x that satisfies both inequalities. We can do this by finding the intersection of the ranges of x determined in both inequalities.

The intersection of -100 < x < 0 and -200 < x < 100 is -100 < x < 0.

Since we are looking for the maximum integer value of x, it is the largest integer less than 0. Therefore, the maximum integer value of x that satisfies both inequalities is x = -1.