# Help with proving identities please

• nevag07
In summary, the conversation discusses proving three different trigonometric identities, including 1/sinxcosx - cosx/sinx = tanx, 1/1+cosx=csc^2x-cscxcotx, and sinx/1-cosx + 1_cosx/sinx = 2cscx. The method of using triangles and the definitions of sine, cosine, and tangent is suggested for proving these identities. It is also mentioned that it is important not to assume anything and to go back to the basic definitions when proving trig identities.
nevag07
i need help with proving these identities
1/sinxcosx - cosx/sinx= tanx

1/1+cosx=csc^2x-cscxcotx

sinx/1-cosx + 1_cosx/sinx = 2cscx

i would really appreciate this

1. $$\frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x} = \frac{\sin x - \sin x \cos^{2} x}{\sin^{2} x \cos x} = \frac{\sin x(1-\cos^{2}x)}{\sin x(\sin x \cos x)} = \frac{\sin x}{\cos x} = \tan x$$2. $$\frac{1}{1+\cos x} = \frac{1}{\sin^{2} x} - \frac{\cos x}{\sin^{2} x} = \frac{\sin^{2} x(1-\cos x)}{\sin^{2}x(\sin^{2}x)} = \frac{1}{1+\cos x}$$you do the last one.

you should come up with $$\frac{2-2\cos x}{\sin x(1-\cos x)}$$ Now factor this, simplify, and see what you get.

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Proof

sine is y/hypotenuse

cosine is x/hypotenuse

tangent is y/x

and X^2 + y^2 = hypotenuse^2

Translate your problems into triangle form and the answers will pop out.

The problem with Courtrigrad's approach (while it is valid), is that he assumes the validity of the identity sin^2 X + cos^2 x = 1. When proving trig identities, it is usually more educational to go back to the basic definitions and not assume anything.

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interested_learner said:

The problem with Courtrigrad's approach (while it is valid), is that he assumes the validity of the identity sin^2 X + cos^2 x = 1. When proving trig identities, it is usually more educational to go back to the basic definitions and not assume anything.

There is nothing wrong with Courtrigrad's method, I don't understand your claim that he assumes anything, I would take
sin2x + cos2x = 1 as a given because it is always true, and does follow from the basic definitions of the trigonometric functions. This is not the case with your method, however, your method to be technical will only work for angles strictly between 0 and 90 degrees and I wouldn't really call your definitions the basic ones.

As long as you don't carry things over and multiplying both sides like you would in algebra (you can find common denominators), that method is fine. The thing to remember is that you need to solve one side until you get to the other.

## What are identities in mathematics?

Identities in mathematics refer to equations or expressions that are true for all values of the variables involved. These equations or expressions are considered to be fundamental and are used to simplify more complex equations or to prove other mathematical relationships.

## Why is proving identities important?

Proving identities is important because it helps to build a stronger understanding of mathematical concepts and relationships. It also allows for the simplification of more complex equations, making them easier to solve.

## What are some strategies for proving identities?

Some strategies for proving identities include using algebraic manipulation, substitution, and using known identities to simplify the given equation. It is also important to break down the equation into smaller parts and work on each part individually.

## Can identities be proven using a calculator?

No, identities cannot be proven using a calculator. Proving identities involves using algebraic techniques and logical reasoning, which cannot be done using a calculator.

## What are some common mistakes to avoid when proving identities?

Some common mistakes to avoid when proving identities include making algebraic errors, failing to simplify both sides of the equation separately, and not using known identities to simplify the equation. It is also important to check the final answer by plugging in values for the variables to ensure that the equation holds true for all values.

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