# Prove that the equation is an identity. State any restrictions.

• goracheski
In summary, the conversation discusses rearranging a mathematical formula and using identities to solve for the left side of the equation. The denominator is correctly factored as (1 - sin(x))(1 + 4sin(x)).
goracheski

## Homework Statement

cos(x)^2/(1+3sin(x)-4sin(x)^2)=(1+sin(x))/(1+4sin(x))

## Homework Equations

We are taking a topic in math where you rearrange one side of the formula to match the other

## The Attempt at a Solution

I have factor 1+3sin(x)-4sin(x)^2 to get (-sin(x)+1)(4sin(x)+1)

if you rewrite 1 + 3 sin x -4 (sin x)^2 using u-sin x you get the quadratic - 4 u^2 + 3u +1 which is fairly easy to factor.

goracheski said:

## Homework Statement

cos(x)^2/(1+3sin(x)-4sin(x)^2)=(1+sin(x))/(1+4sin(x))

## Homework Equations

We are taking a topic in math where you rearrange one side of the formula to match the other

## The Attempt at a Solution

I have factor 1+3sin(x)-4sin(x)^2 to get (-sin(x)+1)(4sin(x)+1)

Continue working on the left side of your equation. What can you do with cos2(x). There's an identity you can use to change it.

The factors you found for the denominator are correct, but it would be better to write them as (1 - sin(x))(1 + 4sin(x)).

## 1. What does it mean for an equation to be an identity?

An identity is an equation that is true for all values of the variable(s) involved. This means that no matter what numbers or expressions are substituted for the variable(s), the equation will still hold true.

## 2. How can I prove that an equation is an identity?

To prove an equation is an identity, you need to show that it is true for all values of the variable(s) involved. This can be done by simplifying both sides of the equation and showing that they are equivalent or by using mathematical properties and rules to manipulate one side of the equation to match the other.

## 3. Are there any restrictions when proving an equation is an identity?

Yes, there may be restrictions when proving an equation is an identity. These restrictions are values that cannot be substituted for the variable(s) in the equation because they would result in undefined or imaginary numbers. It is important to identify and state these restrictions when proving an equation is an identity.

## 4. What are some common restrictions when proving an equation is an identity?

Some common restrictions when proving an equation is an identity include dividing by zero, taking the square root of a negative number, or using logarithms with negative numbers or zero. These actions result in undefined or imaginary numbers, which are not allowed in mathematical equations.

## 5. Why is it important to state restrictions when proving an equation is an identity?

It is important to state restrictions when proving an equation is an identity because it ensures that the equation is true for all valid values of the variable(s). By stating restrictions, you are acknowledging that certain values are not allowed and that the equation may not hold true for them. This helps to avoid any misunderstandings or errors when using the equation in future calculations.

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