# Solving Inequality: (1+tg(x))*cos^2(x) > 1

• Icelove
In summary, the given inequality (1 + tg(x)) * cos^2(x) > 1 can be rewritten as sin(x)cos(x) > sin^2(x) and can be solved using double angle identities from trigonometry.
Icelove

## Homework Statement

(1 + tg(x)) * cos^2(x) > 1, solve the inequality

## Homework Equations

tg(x) = sin(x)/cos(x)
cos(x) * sin(x) = 1/2 sin(2x)

## The Attempt at a Solution

I got my final value of:
1-sin^2(x) + 1/2 sin(2x) > 1
but because of the 2 inside the sin
I'm kinda lost so I can't use quadratic formula I presume

Let me rearrange that a little for you:

1/2sin(2x)>sin^2(x)

which is actually nicer in the following form:

sin(x)cos(x)>sin^2(x).

This should be easier to solve.

grief said:
Let me rearrange that a little for you:

1/2sin(2x)>sin^2(x)

which is actually nicer in the following form:

sin(x)cos(x)>sin^2(x).

This should be easier to solve.

It is important for you to realize where this came from.
If you have notes, cheat sheet, book, or webpage with trig identities, look at the double angle identities.
$$sin2x=2sinxcosx$$

## 1. What is the first step in solving the inequality (1+tg(x))*cos^2(x) > 1?

The first step in solving this inequality is to distribute the cos^2(x) term, which gives us 1 + tg(x)*cos^2(x) > 1.

## 2. Can we divide both sides of the inequality by cos^2(x)?

Yes, since cos^2(x) is always positive, we can divide both sides of the inequality by it without changing the direction of the inequality.

## 3. How do we solve for tg(x)?

To solve for tg(x), we need to isolate it on one side of the inequality. In this case, we can subtract 1 from both sides to get tg(x)*cos^2(x) > 0. Then, we can divide both sides by cos^2(x) to get tg(x) > 0.

## 4. What is the solution set for this inequality?

The solution set for this inequality is all values of x where tg(x) is greater than 0, or in other words, all values of x where the tangent function is positive.

## 5. How can we graph this inequality to visualize the solution?

To graph this inequality, we can plot the points where tg(x) is equal to 0, and then shade the region where tg(x) is positive. This will result in a graph with a vertical asymptote at x = 0, and a shaded region to the right of the asymptote.

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