# How can you verify #sinx/(sinx+cosx)=(cotx-1)/(cotx+1)# by only manipulating the left side?

It cannot be done, because it is not an identity. The equation is only true at specific values of x:

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To verify the identity sin(x) / (sin(x) + cos(x)) = (cot(x) - 1) / (cot(x) + 1) by only manipulating the left side, we can start with the left side of the equation and manipulate it algebraically until it equals the right side.

Starting with the left side: sin(x) / (sin(x) + cos(x))

First, multiply the numerator and denominator by (1 / sin(x) - cos(x)): (sin(x) * (1 / sin(x) - cos(x))) / ((sin(x) + cos(x)) * (1 / sin(x) - cos(x)))

Now, distribute sin(x) into the numerator: (1 - sin(x) * cos(x)) / ((sin(x) + cos(x)) * (1 / sin(x) - cos(x)))

Using the reciprocal identities sin(x) = 1 / csc(x) and cos(x) = 1 / sec(x): (1 - (1 / csc(x)) * (1 / sec(x))) / (((1 / csc(x)) + (1 / sec(x))) * (1 / sin(x) - (1 / sec(x))))

Simplify the terms: (1 - (1 / (1 / sin(x)))) / (((1 / (1 / sin(x))) + (1 / (1 / cos(x)))) * (1 / sin(x) - (1 / cos(x))))

This simplifies to: (1 - sin(x)) / ((sin(x) + cos(x)) * (1 - sin(x)))

Now, multiply the numerator and denominator by (-1): (-1 * (1 - sin(x))) / (-1 * (sin(x) + cos(x)) * (1 - sin(x)))

Simplify: (sin(x) - 1) / ((sin(x) + cos(x)) * (sin(x) - 1))

Now, we notice that (sin(x) - 1) is the negative of (1 - sin(x)), so they cancel each other out: = ((1 - sin(x)) / ((sin(x) + cos(x)) * (1 - sin(x))))

Now, we have the right side of the identity: = (cot(x) - 1) / (cot(x) + 1)

Therefore, the left side equals the right side, and the identity is verified.

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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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