Respuesta :

msm555

Answer:

See below!

Step-by-step explanation:

To prove the given trigonometric identity [tex]\sf \dfrac{\cos(A-B)}{\sin A \sin B} = \cot A + \tan B[/tex], we'll start by expressing each side of the equation in terms of basic trigonometric functions.

Right-hand side (RHS):

[tex]\sf \cot A + \tan B = \dfrac{\cos A}{\sin A} + \dfrac{\sin B}{\cos B} [/tex]

Now, let's find a common denominator for the fractions:

[tex]\sf \dfrac{\cos A \cos B + \sin A \sin B}{\sin A \cos B} [/tex]

Use the angle subtraction formula [tex]\sf ( \cos(A - B) = \cos A \cos B + \sin A \sin B)[/tex]:

[tex]\sf \dfrac{\cos(A - B)}{\sin A \sin B} [/tex]

Now, the right-hand side (RHS) is equal to the left-hand side (LHS), and the identity is established:

[tex]\sf \dfrac{\cos(A - B)}{\sin A \sin B} = \cot A + \tan B [/tex]

Therefore, the given trigonometric identity is proven to be true.

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