# Verifying Trigonometric Identities

• misssue
In summary, the conversation is about a person struggling with verifying trigonometric identities and having trouble with the algebra involved. They are currently working on simplifying the equation cos(x)-tan(x)/sin(x)cos(x) = csc^2 (x) - sec^2 (x) and are unsure of how to proceed after changing it to sin/cos and splitting it into a difference of fractions.
misssue
I have been having lots of trouble verifying trigonometric identities. I know the fundamental identities but I am actually having trouble with the algebra that goes along with the problems.
The problem I am working on now is:

cos(x)-tan(x)/sin(x)cos(x) = csc^2 (x) - sec^2 (x)

(The csc and sec are squared. I didn't know the best way to right that on here)

I tried to change everything to sin/cos but I felt like I made the equation much more confusing doing that.

I got:

(cos(x)/sin(x))-(sin(x)/cos(x))/(sin(x)cos(x)) = (1/sin(x))^2 - (1/cos(x))^2

I am more than a little lost.

I also tried it by changing only the left side and got:

(cot(x)-tan(x)) (1/sin(x)) (1/cos(x))

With either option I don't know where to go next and I'm not even sure if I started correctly.

misssue said:
I have been having lots of trouble verifying trigonometric identities. I know the fundamental identities but I am actually having trouble with the algebra that goes along with the problems.
The problem I am working on now is:

cos(x)-tan(x)/sin(x)cos(x) = csc^2 (x) - sec^2 (x)

(The csc and sec are squared. I didn't know the best way to right that on here)
Looks like you have a typo there, because as written the LHS won't simplify to the RHS. I think you mean this:
$$\frac{\cot x - \tan x}{\sin x \cos x} = \csc^2 x - \sec^2 x$$

(Also, "right" should be "write".)

In which case, why don't you start by splitting the LHS into a difference of 2 fractions and go from there.

I feel silly for writing "write" as "right". There was a typo. Thank you for figuring that out and for the tip!

## 1. What is the purpose of verifying trigonometric identities?

The purpose of verifying trigonometric identities is to ensure that a given equation or expression involving trigonometric functions is true for all values of the variables involved. This is important in mathematics and science because it allows us to simplify and manipulate equations to better understand the relationships between different trigonometric functions.

## 2. How do you verify a trigonometric identity?

To verify a trigonometric identity, you need to use algebraic manipulations and trigonometric identities to transform one side of the equation to match the other side. This involves using properties such as the Pythagorean identities, sum and difference formulas, and double and half angle formulas. You should also keep in mind any restrictions on the variables, such as the domains of the trigonometric functions involved.

## 3. What are some common mistakes to avoid when verifying trigonometric identities?

Some common mistakes to avoid when verifying trigonometric identities include forgetting to distribute negative signs, using the wrong identity or formula, and making algebraic errors such as combining like terms incorrectly. It is important to carefully check each step and make sure it is mathematically sound.

## 4. Can all trigonometric identities be verified?

No, not all trigonometric identities can be verified. There are some identities that are true but cannot be derived from other identities, known as fundamental identities. These include the reciprocal, quotient, and Pythagorean identities. However, most trigonometric identities can be verified using a combination of algebraic and trigonometric manipulations.

## 5. How can verifying trigonometric identities be useful in real-world applications?

Verifying trigonometric identities is useful in real-world applications such as engineering, navigation, and physics. These fields often involve complex equations and relationships between trigonometric functions, and verifying identities allows us to simplify these equations and better understand the underlying principles. This can also help us make predictions and solve problems in practical situations.

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