# How do you prove #[1 - cos(x)]/[sin(x)] = tan(x/2)#?

God bless....I hope the explanation is useful.

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This is a similar process to the other answer,but hopefully this shows a more intuitive approach to determining in what way to manipulate the expressions,

Modifying the right-hand side only,

Using these two identities:

Both squares are cancelled by the square root:

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Here is a solution that does not ignore the sign question in the identities for

Our task now is to

Obviously, if we can show that (1-cos2y)/(sin2y) = sinxy/cosy#, hen we can finish.

It might be too much to keep in our head, so let's write down the choices. Then we'll think about what the next steps would look like and try to make a selection.

OR

OR

Thinking about the next steps, it looks like the middle way might work out best.

It is not necessary to change the variable when we write up the proof, but I think the notation is cleaner and easier to read.

Here's how it looks without changing the variable.

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To prove [1 - cos(x)] / [sin(x)] = tan(x/2), we'll start with the left side of the equation and use trigonometric identities to simplify it.

[1 - cos(x)] / [sin(x)]

= [1 - cos(x)] / [sin(x)] * [1 + cos(x)] / [1 + cos(x)] (Multiplying numerator and denominator by conjugate of sin(x))

= [sin^2(x)] / [sin(x) * (1 + cos(x))] (Using the trigonometric identity sin^2(x) = 1 - cos^2(x))

= sin(x) / (1 + cos(x)) (Cancelling out sin(x))

= (1 - cos(x)) / sin(x) (Rearranging terms)

= tan(x/2) (Using the half-angle identity for tangent: tan(x/2) = (1 - cos(x)) / sin(x)).

Thus, we have proven that [1 - cos(x)] / [sin(x)] = tan(x/2).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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