# Proving this trig identity

• relativitydude
In summary, a trigonometric identity is an equation that is true for all values of the involved variables and is used to simplify and manipulate trigonometric expressions. It is important to prove a trig identity in order to verify its validity and understand the relationship between different trigonometric functions. The general steps to proving a trig identity are to use algebraic manipulations and trigonometric identities to simplify the expression and compare it to the other side of the equation. Some common trig identities that are useful in proving identities include Pythagorean identities, double angle identities, sum and difference identities, and half angle identities. Practicing with different examples and seeking assistance can help improve skills in proving trig identities.

#### relativitydude

This is a tricky one, sin(3x)cos(x)=sin(x)cos(x)(3-sin^2(x))

I have tried addition formulas and expanding sin(3x) into something else, any help?

That's not just "tricky", it's false! Try $$x= \frac{\pi}{4}$$.

Like Halls said, remember that an identity is an equation that all values are solutions.

It should have been

$$\cos x\sin 3x\equiv \cos x\sin x\left(3-4\sin^{2}x\right)$$

Daniel.

## What is a trig identity?

A trigonometric identity is a mathematical equation that is true for all values of the variables involved. It is used to simplify and manipulate trigonometric expressions.

## Why is it important to prove a trig identity?

Proving a trig identity allows us to verify its validity and understand how different trigonometric functions are related to each other. It also helps in solving more complex trigonometric equations.

## What are the steps to proving a trig identity?

The general steps to proving a trig identity are:
1. Start with one side of the equation and use algebraic manipulations to transform it into the other side.
2. Use trigonometric identities and formulas to simplify the expression.
3. If needed, use basic algebraic principles to further simplify the expression.
4. Compare the simplified expression to the other side of the equation to see if they are equivalent.
5. If the two sides are equal, the identity is proven. If not, continue to manipulate the expression until it is equivalent to the other side.

## What are some common trig identities that are useful in proving identities?

Some common trig identities include:
- Pythagorean identities: sin^2(x) + cos^2(x) = 1 and tan^2(x) + 1 = sec^2(x)
- Double angle identities: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos^2(x) - sin^2(x)
- Sum and difference identities: sin(x+y) = sin(x)cos(y) + cos(x)sin(y) and cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
- Half angle identities: sin(x/2) = ±√[(1-cos(x))/2] and cos(x/2) = ±√[(1+cos(x))/2]

## How can I practice and improve my skills in proving trig identities?

The best way to practice and improve your skills in proving trig identities is by attempting different examples and problems. You can find practice problems in textbooks, online resources, or you can create your own equations to solve. It is also helpful to review and understand common trig identities and their derivations. Additionally, seeking assistance from a tutor or joining a study group can also aid in improving your skills.