# Verify: #-(cotA+cotB)/(cotA-cotB) = sin(A+B)/sin(A-B)# ?

Expand and simplify RHS then apply identities for

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To verify the equation ( -\frac{\cot A + \cot B}{\cot A - \cot B} = \frac{\sin(A + B)}{\sin(A - B)} ), we'll start by expressing cotangents in terms of sines and cosines:

[ \cot A = \frac{\cos A}{\sin A} ] [ \cot B = \frac{\cos B}{\sin B} ]

Substitute these expressions into the left-hand side of the equation:

[ -\frac{\frac{\cos A}{\sin A} + \frac{\cos B}{\sin B}}{\frac{\cos A}{\sin A} - \frac{\cos B}{\sin B}} ]

Simplify this expression by finding a common denominator:

[ -\frac{(\cos A \sin B + \cos B \sin A)}{(\cos A \sin B - \cos B \sin A)} ]

Now, use the sum and difference identities for sine:

[ \sin(A + B) = \sin A \cos B + \cos A \sin B ] [ \sin(A - B) = \sin A \cos B - \cos A \sin B ]

Substitute these identities into the expression:

[ -\frac{\sin(A + B)}{\sin(A - B)} ]

This matches the right-hand side of the equation. Therefore, the equation 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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