# Proving the Trig Identity tanx+cotx=2csc2x

• DethRose
In summary, you are trying to solve for x by multiplying tan(x) + cot(x) by tan(x) and proceeding from there. You should be able to get the LHS in terms of sin x and cos x .
DethRose
Hey I've got an assignment on trig identities and can't figure this one out.

Prove the Identity:

tanx+cotx=2csc2x

I got to

tanx+cotx= 1/2 sin2x=1/4sinx2cosx

but when i get to the point where i have numbers in front of the sinx or cosx i don't know what to do.

Thanks for any help

try multiplying tan(x) + cot(x) by tan (x) and proceeding from there. I'll start you off,

$$tan(x) + cot(x) = \frac{sec^2(x)} {tan(x)}$$

Write the LHS in terms of $$\sin x$$ and $$\cos x$$. You should get:

$$\frac{1}{\sin x \cos x} = \frac{1}{\frac{1}{2}\sin 2x} = \frac{2}{\sin 2x} = 2\csc 2x$$

Last edited:
im sorry but i still have no idea how to do this question

have you tried changing everything to sin and cos ? doing that you will be able to get the right side to equal the left.

if you're still stuck just post what you've tried to do and someone could point out where you went wrong or give you some advice on what the next step would be

You know that $$\tan x = \frac{\sin x}{\cos x}$$ and $$\cot x = \frac{\cos x}{\sin x}$$Therefore:$$\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^{2} x + \cos^{2} x}{\sin x \cos x}$$.

$$\sin^{2} x + \cos^{2} x = 1$$ so we have $$\frac{1}{\sin x \cos x}$$.

Now we know the identity $$\sin 2x = 2\sin x \cos x$$. Thus $$\sin x \cos x = \frac{1}{2}\sin 2x$$.

So now we have $$\frac{1}{\frac{1}{2}\sin 2x}$$. Remembering that $$\frac{1}{\sin x} = \csc x$$ we can deduce that $$\frac{1}{\frac{1}{2}\sin 2x} = 2\csc 2x$$

Last edited:
isnt it supposed to equal 2csc2x not 2cscx?

DethRose said:
isnt it supposed to equal 2csc2x not 2cscx?

It does he probably made a typo. There are loads of ways to do it but basically all you have to do is find a page full of trig identities and fiddle about.

Is it incorrect to work both sides of the problem and meet in the middle, or do you need to get one side to completely equal another, because I can easily get both sides to be 1/sinxcosx.

You have to get one side completely equal to the other. Working both sides of the problem and "meeting in the middle" is not correct.

And that is because in order to meet in the middle, your assuming the 2 sides are infact equal. If you mean change one side with identities, and change the other, and make them the same, then from there you can work backwards and get it from one side anyway.

OK this is a easy problem

L.S tanx+cotx
=Sinx/cosx+cosx/sinx

= sinx^2+cos^2/cosxsinx

=1/cosxsinx

R.S 2CSC2X
=2*1/2sinxcosx (The 2's here cancel out)
=1/sinxcosx

L.S=R.S

I tried to be as clear as possible, sorry but I'm not used to doing math on the computer :(

## 1. What is a trigonometric identity proof problem?

A trigonometric identity proof problem is a mathematical problem that involves proving the equality of two trigonometric expressions using the basic trigonometric identities (such as the Pythagorean identity, double angle identities, etc).

## 2. How do I approach a trigonometric identity proof problem?

The key to solving a trigonometric identity proof problem is to use the basic trigonometric identities and algebraic manipulations to transform one expression into the other. It may also be helpful to draw a diagram or use trigonometric properties, such as even/odd identities, to simplify the expressions.

## 3. What are the common mistakes to avoid in a trigonometric identity proof problem?

Some common mistakes to avoid in a trigonometric identity proof problem include forgetting to use the basic identities, making algebraic errors, and not simplifying the expressions enough. It is also important to check that both sides of the equation are equivalent at each step of the proof.

## 4. How can I practice and improve my skills in solving trigonometric identity proof problems?

One way to practice and improve your skills in solving trigonometric identity proof problems is to work on a variety of problems, starting with simpler ones and gradually increasing in difficulty. You can also find online resources or textbooks with practice problems, and seek help from a teacher or tutor if needed.

## 5. Are there any tips or tricks for solving trigonometric identity proof problems?

Some helpful tips for solving trigonometric identity proof problems include checking if the expressions can be simplified using trigonometric properties, using symmetry to your advantage, and breaking down the problem into smaller, more manageable steps. It is also important to stay organized and clearly write out each step of the proof.

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